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How harsh will winter be? Six organizations issue forecasts.
A look at forecasts from NOAA, AccuWeather, Weather.com and others
By Zach Rosenthal
October 13, 2022 at 10:42 a.m. EDT
Mary Molitor passes Lincoln Park in D.C. on March 12, 2022. (Amanda Andrade-Rhoades/For The Washington Post)
While it is still several weeks until the official start of winter on Dec. 21, several organizations are already unveiling their nationwide Winter 2022-2023 forecasts.
Note that even the most scientifically advanced seasonal outlooks cannot pinpoint what the weather will be in a particular place at a particular time this far in advance. But, with varying levels of success, they can paint a broad picture of how hot or cold or wet or dry different parts of the country may be compared to average.
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Winter's coming: Here's what to know about long-range weather outlooks
Among the winter outlooks issued by meteorologists so far, most agree that the southern United States will be drier and warmer than normal, with the best chance of colder and stormier-than-normal conditions in the northern tier, Midwest and Ohio Valley. Such projections reflect typical conditions that develop during La Niña events, which are associated with an episodic cooling of ocean waters in the tropical Pacific. This year, La Niña is forecast to prevail for a third straight winter.
Whether you are a fan of snow, here is the latest roundup of what meteorologists are saying about the weather in the coming months. For entertainment purposes, we also summarize the outlooks from the Farmers' Almanac and its rival, the Old Farmer's almanac — but meteorologists put little stock in those predictions.
NOAA's latest forecast
Although the official winter forecast from the National Oceanic and Atmospheric Administration will not come out for another few weeks, the agency's Climate Prediction Center does issue official outlooks for temperature and precipitation up to 13 months in advance.
For the first three months when winterlike conditions begin in earnest — November, December and January — abnormal cold is not expected anywhere in the country. A large swath of the country, from the East Coast down into the Sun Belt and into the Mountain West, is projected to experience above-normal temperatures, with the highest probability of abnormal warmth in Arizona and New Mexico.
From February to April, above-normal temperatures are projected to continue along the East Coast, in the Southeast and into the Southwest, with the greatest chance of warmer weather along the Mid-Atlantic and Southeast coastlines as well as parts of the Southwest. Below-average temperatures are forecast in the northern contiguous United States, stretching from northern Michigan to northern Washington state.
Precipitation-wise, the period from November to January is expected to bring below-average precipitation — and thus diminished early-season snow and rain chances — in much of the southern half of the country, with the greatest chances of below-normal precipitation forecast from coastal South Carolina and Florida all the way to the shores of far Southern California. Above-normal precipitation is possible in northwestern Montana, northern Idaho and northeastern Washington state.
From February to April, below-normal precipitation is forecast in the Southwest and coastal portions of the Southeast, but areas including Texas may see a respite from less-than-normal snowfall and rainfall. Above-normal precipitation is forecast in part of the Ohio Valley, an area that could pick up above-normal snowfall if temperatures remain low enough.
AccuWeather's winter forecast
AccuWeather's official 2022-2023 U.S. winter forecast is rather bleak for snow lovers. AccuWeather senior meteorologist Paul Pastelok and his team say that this winter's setup is complicated by several other factors — including the Hunga Tonga volcanic eruption in the early days of 2022.
La Niña could enter rare third straight year. Here's what that means.
AccuWeather forecasters are predicting a more active severe thunderstorm season in the southeastern states during the winter months because of warmer-than-normal ocean temperatures.
Pastelok said that the warmer ocean temperatures could help to fuel a "potentially big system" that could affect the East Coast in the latter half of winter. But in general, AccuWeather is predicting a season of less snowfall on the Eastern Seaboard.
While AccuWeather forecasts that snowfall will be suppressed, the company does not necessarily expect overall precipitation to be below normal as well, with milder temperatures leading to several all-rain events this winter. Those rainstorms may cause flooding in the Ohio Valley and along the Mississippi River, AccuWeather says.
During the back half of the winter, AccuWeather says, colder conditions finally will enter the country and drop cold air into the central United States, bringing heavy snow to parts of the central Plains and the Rocky Mountains. In the West, generally dry conditions will do little to ease the region's persistent drought.
How will the Hunga Tonga volcanic eruption play into the forecast? AccuWeather says that the lingering water vapor in the atmosphere from the eruption could cause a warmer winter than normal but that the magnitude of the effect is unknown.
Weather.com's winter outlook
Weather.com's official winter outlook — like NOAA's and AccuWeather's — calls for above-normal temperatures in the South, while far-northern parts of the continental United States manage to stay below average, these conditions being driven by La Niña.
December is favored to be the chilliest month on the Eastern Seaboard, with lower-than-normal temperatures expected from the Great Lakes down into parts of the Mid-Atlantic and Northeast. On the other side of the country, temperatures in the Southwest and the Rockies are expected to be well above average.
By January, most of the country is mild, with lower temperatures farther north and a serious chill entering the Upper Midwest and Great Lakes region by February, while the Southeast — especially toward Florida, warms up.
WeatherBell's winter forecast
The winter outlook from WeatherBell Analytics, a weather consulting firm, should put more pep in the step of snow lovers. The firm predicts temperatures that are normal to slightly below normal for nearly all of the country from November 2022 to March 2023.
The lowest temperatures are expected in the Ohio Valley into the Upper Midwest, which are forecast to be 1 to 3 degrees below normal.
Higher temperatures are forecast for the West; WeatherBell projects temperatures 1 to 3 degrees above normal in the Southwest. That part of the country also is expected to receive less snow than normal. That forecast extends to some popular California ski resorts.
WeatherBell is forecasting slightly above-normal snowfall, 125 percent of the seasonal average, in the Midwest, through the Ohio Valley and into the interior Northeast. It calls for snowfall to be above normal toward the East Coast as well.
Farmers' Almanac forecast
The Farmers' Almanac winter snow forecast is predicting an early start to winter, with a cold and stormy December. That storminess is not expected to slow in the eastern half of the country, with the almanac suggesting snowy conditions into the Northeast. Along the Interstate 95 corridor, which often is the rain-snow line for major storms, the Farmers' Almanac suggests more snow than rain.
In the Southeast, the Farmers' Almanac's expected East Coast storms are more likely to bring rain, although chilly conditions that enter the region in January could cause wintry precipitation there, as well. The Farmers' Almanac says temperatures in the Southeast and Northeast should become milder in February, though.
The Farmers' Almanac says conditions in the Upper Midwest will be "glacial," and it suggests there will be plenty of snow and chilly conditions for winter lovers to enjoy — including the potential for a White Christmas. In mid-January, the Farmers' Almanac says, temperatures could drop as low as 40 degrees below zero in parts of the region.
The Farmers' Almanac predicts that across the region, March will go out like a lion, with a variety of conditions, including heavy snowfalls, heavy rain and gusty thunderstorms.
Old Farmer's Almanac forecast
The Old Farmer's Almanac is predicting a divided nation — with harsh winter in the East and mild weather in the West. Out West, even if temperatures are mild, the predicted above-average precipitation can help to ease the region's drought.
Let's turn the Farmer's Almanac into something real — and useful
In the East, the almanac predicts above-average snowfall for a vast area, from North Carolina to central New England to the Ohio and Tennessee Valleys, as well as the Great Plains.
The almanac forecasts cold and wet conditions down even into Florida, with the worst of the cold forecast for January. It says that temperatures there could drop 4 degrees below normal for the month — something that could result in a damaged citrus crop and stunned iguanas.
Washington, D.C., Snow Tracker
0"Current total snowfall
Average season-to-date snow 6.5"
Average seasonal snow 13.7"
Last year's snow 13.2"
Record most snow 56.1"
Record least snow 0.1"
See the winter outlook
Looking for inspiration? Try these recipes to prepare dishes with confidence.
A hearty lentil soup that's easy on your budget
This rich stew from a Native American chef connects us to the land
Crunchy stuffed sweet pepper 'boats' are a family-friendly meal
Tweets by @capitalweather


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Maury Blackman
Maury Blackman has led high-growth tech companies for more than 25 years. As CEO of Premise, Maury is passionate about the decision-making power that comes when highly engaged communities and leading-edge technology are combined. Under Maury's leadership, Premise was recognized as one of Deloitte's Fast 500 in 2020, and has grown into the world's foremost crowdsourced insights solution.
Prior to Premise, Maury was CEO of Accela, a civic-tech start-up designing and delivering government software to improve efficiency and citizen engagement. In 2016, EY awarded him the prestigious Northern California's Entrepreneur of the Year award for his work as Accela's Chairman and CEO. Accela was sold to Berkshire Partners in 2017. Following graduation from the University of Houston, Maury served as a commissioned field artillery officer in the United States Army. He's also an active investor, board member and advisor to many notable technology companies.
Over the past 15 years, Maury has participated in over 100 endurance events, including marathons, trail running, mountain biking, open water swims, cycling, and triathlons. His primary focus is Ironman distance triathlons (2.4m swim, 112m bike, 26.2 run), completing 11 full and 20+ half Ironman races.
Maury is particularly passionate about infrastructure reform, and you can read about his work in the space here.

redpajama_set_name: RedPajamaCommonCrawl


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Bechara, C, Pallerla M, Zaltsman Y, Burlina F, Alves ID, Lequin O, Sagan S. 2013. Tryptophan within basic peptide sequences triggers glycosaminoglycan-dependent endocytosis. The FASEB Journal. 27(2):738-749.
Walrant, A, Vogel A, Correia I, Lequin O, Olausson BES, Desbat B, Sagan S, Alves ID. 2012. Membrane interactions of two arginine-rich peptides with different cell internalization capacities. Biochimica et Biophysica Acta (BBA) - Biomembranes. 1818(7):1755-1763.
Sebai, SC, Milioni D, Walrant A, Alves ID, Sagan S, Huin C, Auvray L, Massotte D, Cribier S, Tribet C. 2012. Photocontrol of the Translocation of Molecules, Peptides, and Quantum Dots through Cell and Lipid Membranes Doped with Azobenzene Copolymers. Angewandte Chemie International Edition. 51(9):2132-2136.
Walrant, A, Bechara C, Alves ID, Sagan S. 2012. Molecular partners for interaction and cell internalization of cell-penetrating peptides: how identical are they? Nanomedicine. 7(1):133-143.
Walrant, A, Correia I, Jiao C-Y, Lequin O, Bent EH, Goasdoué N, Lacombe C, Chassaing G, Sagan S, Alves ID. 2011. Different membrane behaviour and cellular uptake of three basic arginine-rich peptides. Biochimica et Biophysica Acta (BBA) - Biomembranes. 1808(1):382-393.
Bechara, C, Jiao CY, Burlina F, Alves ID, Chassaing G, Sagan S. 2010. Role of cell-surface carbohydrates and plasma membrane components in the internalization of cell-penetrating peptides. Drug Discovery Today. 15(23-24):A58.


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Grant number like: CH-50357-07
Virginia Museum of Fine Arts (Richmond, VA 23220-4007)
Suzanne H. Freeman (Project Director: November 2005 to January 2011)
Robin Nicholson (Project Director: January 2011 to April 2011)
Lee Bagby Viverette (Project Director: April 2011 to July 2011)
CH-50357-07
$610,000 (approved)
$610,000 (offered)
$610,000 (awarded)
Endowment for new Virginia Museum of Fine Arts Library
Endowment for the Head Librarian, a reference librarian, acquisitions, equipment, furnishings, and a professional development and travel fund.
VMFA seeks $800,000 to be matched 4:1 for a total of $4MM to provide endowment for a new art research Library to be located in the Museum's new wing scheduled to open in 2007-08. The new Library will offer extensive humanities holdings including 132,500 volumes. Endowment will support the Head Librarian and one Librarian position, acquisitions, preservation equipment and materials, furnishings, and a professional development and travel fund. The Library is central to the Museum's commitment to provide a humanities framework for interpreting all of its collections, exhibitions, publications, and educational programs.


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LG's New Mobile Keyboard Rolls Up, Fits in Your Pocket
The Bluetooth-connected Rolly Keyboard folds up to create a "stick" that fits into a pocket.
By Angela Moscaritolo
https://www.pcmag.com/news/lgs-new-mobile-keyboard-rolls-up-fits-in-your-pocket
Is the keyboard on your smartphone or tablet slowing you down? LG has a solution.
The company on Thursday introduced what it calls "the industry's first solid rollable wireless portable keyboard." The Rolly Keyboard folds up along its four rows to "create an easy-to-carry 'stick' that fits into one's pocket as easily as any purse or briefcase," LG said.
Featuring "high-contrast keys" and a fold-out stand, LG said the keyboard makes typing "extremely comfortable" thanks to its 17mm key pitch, which rivals the 18mm key pitch found on most desktop keyboards.
It's made of impact-resistant, durable polycarbonate and ABS plastic, which offers "satisfying tactile feedback" you won't get from other flexible keyboards made of silicone, LG said. Even better — to connect it to your smartphone or tablet, all you need to do is unfold the Rolly Keyboard. This will enable the auto-pairing function, letting you connect to up to two different devices at the same time via Bluetooth 3.0. You can toggle between the two devices with a simple key press.
The Rolly takes just one AAA battery, which should give it enough juice for up to three months of "average use," LG said.
"LG Rolly Keyboard is just one of the many premium input devices we'll be unveiling in the coming months as we expand our accessories offerings," Seo Young-jae, LG's vice president in charge of innovative personal devices, said in a statement. "The goal was to create a product that could add more value to LG smartphones and tablets at the same time offering a unique design proposition that hadn't been explored before."
LG plans to show off the Rolly Keyboard at the IFA trade show in Berlin next week.
At this point, there's no word as to how much it will set you back, but LG promised pricing details at launch. The Rolly Keyboard will be available in the U.S. this September before arriving in Europe, Latin America, and Asia in the fourth quarter.
Tiny 59-Key Keyboard Created Using a Raspberry Pi Pico
By Matthew Humphries
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By Mike Epstein
About Angela Moscaritolo
Angela Moscaritolo is PCMag's expert on fitness and smart home products. Teaching yoga for nearly 10 years has helped her in evaluating a range of connected health products including fitness trackers, heart rate monitors, rowers, smart scales, stationary bikes, strength training machines, treadmills, and more. Angela also tests smart home devices from her ranch in Florida, including air purifiers, kitchen appliances, and robot vacuums, just to name a few.
To date, Angela has written more than 6,000 articles and reviews for PCMag. Before coming into her analyst role in 2020, she spent eight years as a PCMag reporter covering consumer tech news. Prior to joining the PCMag team, she was a reporter for SC Magazine, focusing on hackers and computer security.
Angela is a graduate of West Virginia University's Perley Isaac Reed School of Journalism, and started her career writing for newspapers in New Jersey, Pennsylvania, and West Virginia.
Read the latest from Angela Moscaritolo
What Does Big Tech Know About You? Basically Everything
NordicTrack Launches First Ever Amazon Alexa-Compatible, Voice-Controlled Dumbbells
CES 2022 Envisions Your High-Tech House: The 10 Best Smart Home Devices of the Show
The Best Sleep Tech for Catching Your ZZZs
Peloton's $495 Guide Camera Uses AI to Help You Strength Train Smarter
More from Angela Moscaritolo


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Nadra to build data exchange layer for govt agencies
Tahir Amin Published September 27, 2022
ISLAMABAD: The National Database and Registration Authority (Nadra) will build a data exchange layer for government agencies to exchange information securely, as well as, provide citizens with a digital "wallet" to secure credentials, it is learnt.
Official documents revealed that it is the part of Stakeholder Engagement Plan (SEP) under the World Bank financed "Digital Economy Enhancement Project (DEEP)" worth $200 million.
The documents noted that the DEEP will support activities that will increase access to and promote the use of digital services across Pakistan, with a focus on broadband access and reducing the significant digital divide currently existing in many parts of the country.
The project will also support cross-cutting improvements in the enabling environment—supporting reforms to expand broadband connectivity and to strengthen the policy and legal framework for the digital economy.
DEEP will also enable institutional, policy, and regulatory global best practices through technical assistance to the implementation agencies—including the MoITT, Nadra, Punjab IT Board, Board of Investment (BoI), and other provincial institutions. Under the project, Nadra will build a data exchange layer for government agencies to exchange information securely, as well as provide citizens with a digital 'wallet' to secure credentials.
Non-filers: NADRA re-working on data for correction of profiles
The project is aligned with the 2018 Digital Pakistan Policy, and in particular objective, which calls for the establishment of a holistic (government wide) enterprise architecture and the integration of government databases and systems through e-government service portals.
It is also aligned with the World Bank Group's (WBG) Country Partnership Strategy and its objective to catalyze inclusive and sustained growth. It also corresponds with the Systematic Country Diagnostic's (2020) identified priority of "levelling the playing field" by investing in policies to improve competitiveness and promote equity and inclusion. It will improve the efficiency of digitally enabled public services. These activities are expected to improve access to and quality of basic services, thereby improving social and human development outcomes in Pakistan.
The project is comprised of three components:
Component 1: Improving digital economy, governance and service capabilities. This component will provide technical assistance to the Ministry of Information Technology and Telecommunications (MoITT) for policy and regulatory reforms to address regulatory and market failures in the country's connectivity infrastructure.
Component 2: Pakistan Business Portal. This component will support the Board of Investment (BoI) to modernize regulatory regimes in Pakistan at three levels of government: federal, provincial, and municipal. The first stage entails reviewing, mapping, and developing a catalogue of registrations, certificates, licenses, and others (RLCOs) across the three levels of the government, potentially including up to 800 government agencies relevant to dealing with investing and operating businesses in Pakistan.
Component 3: Project management. This component will cover the establishment of Project Management Units (PMU).
World Bank nadra Digital Economy Enhancement Project Digital Pakistan Policy Project Management Units govt agencies
Shujaat to remain PML-Q president, rules ECP


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//Rather Be
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Jack Patterson, Luke Patterson, Grace Chatto, and Neil Amin-Smith make up the electro-pop group, Clean Bandit. Last year, their chart-topping single "Rather Be", a collaboration with Jess Glynne remained at the top of the chart for four weeks. They went on to win the...
With Her Debut EP Coming Out Next Year, The Promising Singer-Songwriter, Emma Kate Chats About Her Beginnings and Her Bright Musical Future!
Emma Kate is a very promising singer-songwriter with a debut EP hopefully out early next year. This collection features her new catchy debut single "Neon Yellow Hearts". The single was co-written by Aaron Kamin from The Calling. Emma Kate's "Neon Yellow Hearts"...


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Jackson's, California Country Park
On July 22, 2016 August 22, 2017 By ediblereadingIn Reviews
Jackson's stopped evening opening on 3rd September 2016. I've left the review up for posterity.
I've never quite worked out what the limits of the ER catchment area are, but Jackson's has to be right on the edge. It's only thirty minutes out of town but somehow it feels like a longer, more circuitous drive than that, out through Shinfield and past Arborfield before the countryside gets quite beautiful and the houses start to thin out. Then you go along Nine Mile Ride, turn off into California Country Park and suddenly it's like being in Canada (or at least Center Parcs), all tall trees and wooden signs, Lycra-clad cyclists and healthy-looking types: types, in fact, unlike me.
The sun still flecked the leaves on a July evening as I made my way to the restaurant but I could completely see what it would be like during the daytime, full of families enjoying the great outdoors and the aforementioned cyclists taking a pit stop before heading on towards somewhere suitably pastoral. Jackson's – next to the oddly named California Lake – tries to offer something at all times, so they do breakfast (all day, apparently: worth knowing) and sandwiches, burgers and jacket potatoes for lunch before morphing into a restaurant in the evening. And a Proper Restaurant at that, if the menu is to be believed, full of little touches and hints that make you really want to turn up and order everything.
The dining room is a big space full of chunky, solid, simple wooden furniture with white and red tablecloths. I assume the only things that change from day to night are the tablecloths and the mood lighting, little lamps sitting on rough-hewn rustic tables, but I imagine the space must be lovely at any time of day. I caught sight of the wood-burning stove and, for the first time this summer, found myself wishing it was ever so slightly darker and colder. Only two other tables were occupied, which was a tad awkward; they looked delighted that somebody had joined them.
I said the menu was full of little touches. Have a look at how they described our first starter if you don't believe me: "Nutmeg and Thyme Confit Chicken Thigh, Chicken Broth, Crisp Skin, Wild Mushrooms, Broad Beans, Charred Corn & Spinach Foam". Sounds good, don't you think? And generally speaking it was. The thigh meat, soaked in the broth, was like the wonderful bit at the end of a roast dinner when you get to the last shreds of chicken, falling apart, soaked in rich salty gravy. Intensely good. The wild mushrooms were served as little splodges of purée that added creaminess but were otherwise a little lost. The broad beans – very large, very green, beautifully fresh – were top-notch. Only some of the corn was actually charred, but it added a nutty, chewy texture, nonetheless. No crisp skin, either, which is a great shame as they're always two welcome words on any menu I look at. Last of all, the spinach foam added great colour (thankfully not too reminiscent of frogspawn) but not a lot else.
So, lots going on but did it justify all that complexity? No, not really. I could admire the creativity and all those – what's the word they use on Masterchef? – processes, but I'd have been just as happy with a big ramekin of chicken, broth and broad beans, especially as the dish was too shallow for me to scoop up the remaining broth at the end.
The other starter – "Sweet Pickled Heritage Carrots, Truffle Mayo, Poached Local Hens Egg, Summer Truffle, Parmesan & Honey Dressing" was more successful. A pretty, delicate, deceptively substantial dish it was built around those glorious spirals of sweet, crunchy pickled carrot and a mayo which managed to get those truffle notes just the right side of overpowering (although I do have a very high capacity for truffle). The parmesan shavings added grit and salt and the egg, poached just right, oozed enough golden yolk to bring everything together nicely. There were also pea shoots, which I could have done without (given the kitchen sink approach to the menu I'm surprised these weren't mentioned) and little blobs of insubstantial dressing, also dotted with tiny pieces of truffle. Pickle, truffle, parmesan, poached egg: these are a few of my favourite things, and I loved this dish.
After two thumbs up for the starters I felt hopeful for the main event, and I was right. Well, half right: the confit duck leg was lovely. It was salty and rich with the meat falling away from the bone, exactly as it should be. The skin was perfectly crisp (nearly compensating for that chicken skin that went MIA earlier in the meal) with no flabby edges, almost how I think pork scratchings should be but so rarely are.
So far, so simple, but underneath the duck was where things got a lot more interesting; a pile of "young" peas (whatever they are, petits pois presumably) was given freshness and tang with fresh mint and crumbled feta. Some of the peas had also been puréed and served in pretty acid-green dollops which were surprisingly rich when smeared over a piece of the duck leg. The ham hock and potato terrine – a brick of potato with a layer of ham hock pressed as a seam through the middle – wasn't such a hit. The potatoes were too hard and too lukewarm to be enjoyable and I ended up pulling out the ham hock to eat on its own; far from a punishment but probably not what the kitchen intended.
Luckily, we'd ordered truffle chips on the side and my goodness, they were cracking: thick-cut, fluffy inside, tossed in parmesan and then sprinkled with what I imagine was truffle oil. But there were also dark speckles of truffle throughout the bowl and the whole thing gave off that distinctive earthy, dirty aroma (oh to be a truffle pig!). Five pounds for these, but worth the money and possibly worth the price of admission alone. If I was being critical they could have done with being a little crispier, but at the end there was a little tangle of truffly molten cheese which I scooped up with my fork and at that point, such minor quibbles faded into the background.
If they were the kitchen at its best the other main was the biggest dud of the visit. I wanted to try something more straightforward to test the range of the menu so I went for the chicken breast burger and here's where things went awry. It was described as coming with parmesan, truffle mayo and double smoked bacon. Well, the bacon may have been smoked twice but it was put on my burger zero times, something I didn't realise until it was too late to send it back. The chicken itself was lovely, slightly flattened and breaded, almost like a chicken Milanese, but the outside – bit of a theme here – wasn't crunchy and crispy as it should have been. The Parmesan was a thin layer, almost more like a sauce than a discernible slice of cheese and that was good, although it made things rather soggy. No truffle mayo that I detected either, unless my tastebuds had been numbed to truffle by then. I suppose it's possible. Nice brioche though, and the usual suspects – tomato, iceberg, right ahead, red onion – were joined by a row of little crunchy cornichons which almost redeemed matters. But really, this burger cost fourteen pounds and it just wasn't worth that.
I do need to single out the triple cooked chips which came with the burger for two reasons. One is that they came in a Jenga stack. Now, some of you may not remember this but Jenga chips were the "food on slates" of their day. Everyone hated them, everyone railed against them and as a result I thought they had become extinct. I was so surprised to see them on a plate in a restaurant like this that I wanted to check the date on my phone to see if some kind of Quantum Leap oddness had gone on. But that aside, they were a sad and flabby bunch. Maybe it was the time taken painstakingly arranging them in a tower, maybe they just weren't very good to start with, but they weren't triple cooked. They were somewhere between once and twice cooked, I decided, before eating a couple which I would describe as barely cooked chips.
The dessert menu was full of interesting things but only one of us fancied one (that chicken burger had somewhat dampened enthusiasm). The first item on there really intrigued me, and when I asked for advice it was also the one recommended by the serving staff and that, in restaurant terms, is kismet, right there. Liquorice panna cotta with rhubarb and stem ginger might have been the dish of the whole meal. Yes, liquorice panna cotta. If you've ever tried the basil and balsamic panna cotta at Pepe Sale you'll know that it's worth taking a risk sometimes with flavours that don't obviously go and this was definitely true here, especially after some of the inventive touches in the previous courses.
The panna cotta itself was dreamy – super thick, well set and extremely creamy. It had most delicate liquorice flavour to it, almost metallic in the mouth, nearly an aftertaste, the hint of something. I'm lost in a reverie remembering it, and clearly struggling to adequately describe even if you don't like liquorice – and I'm not a massive fan – I'd still thoroughly recommend it. It came with an embarrassment of riches: dots of ginger and rhubarb purée; sweet, intense pieces of cooked rhubarb, crumbled stem ginger cookie and a scoop of rhubarb ice which somehow managed to be halfway between a sorbet (fresh and bright) and an ice cream (smooth and creamy). I don't know how the chef came up with the idea to put those flavours together and I'm not sure I care. It doesn't really matter: I just loved it.
Sadly I was driving so I only got to try one glass of wine; the carignan was a nice, rich red (good enough for my companion to have it as their second glass). I'm told that the Riesling was nice, fresh and not as good as the carignan. That was after some grumbling about the chicken burger, so I managed to be on the receiving end of both food envy and wine envy. Oh, and I had a diet Coke because after the glass of wine I wasn't feeling particularly imaginative. Still, it came in a glass bottle rather than out of a syphon, and somehow that always feels like proper Coke to me.
Service was pretty good; the maitre d' was charming and effusive and the other, younger staff, although not quite so engaging, also did a good job. Everything was nicely timed, too, which could so easily not have been the case on such a quiet weekday night: nothing came too quickly or too slowly, plates were cleared away when they should be and so on. Two starters, two mains and one dessert plus three glasses of wine and a soft drink came to seventy-three pounds, excluding tip. The starters hover around the eight pound mark and most of the mains are under fifteen quid. For food this imaginative, that's pretty impressive.
What makes this gig, for me, is when I discover a gem. Ideally somewhere in the centre of town, although that gets increasingly difficult, but in any case somewhere wonderful – preferably independent – that you won't have heard of, offering imaginative, fairly priced food. Jackson's should be that, and it so nearly is. So why isn't it? The food, by and large, and the service do indeed say that you're in a Proper Restaurant. But the elephant in the room is the room itself: the owners have tried to make a step change from day to night but I couldn't shake the feeling that I was having dinner in a café.
I know this shouldn't matter: I've always said that if the food is good none of the rest matters, so I'm troubled that in this case I find it does. Perhaps it's because of the half hour drive. Perhaps it's because the room was empty and lacked ambience. Maybe I would have felt differently if it had been full, on a cold winter night with the woodburning stove glowing and the table lamps shining, having dinner with other happy strangers in this little cabin in the woods. So a qualified recommendation from me: if none of that bothers you please go, fill up the restaurant, enjoy some really clever food at simple prices and keep them busy. Personally, I'll go back when the clocks do.
Jackson's – 7.3
California Country Park, Nine Mile Ride, Finchampstead, RG40 4HT
http://jacksonscalifornialake.co.uk/
burgersCalifornia Country ParkchickenclosedJackson'sReadingrestaurantsServicewineWokingham
Sam's Wraps
The last post


text:
\section{Introduction}\label{sec_1}
	In 1601, Kepler obtained a detailed set of observations of the motion
	of the planet Mars from the Danish astronomer Tycho Brahe \cite{Brac}.
	From his analysis of these data, Kepler determined that the path of
	Mars is an ellipse, with the sun located at a focal point, and that
	the radius vector from the sun to the planet sweeps out equal areas in
	equal times.  The direct problem was to determinate the nature of the
	force required to maintain elliptical motion about a focal force
	center.  This direct problem remained unsolved until after 1679, when
	Newton determined the functional dependence on distance of the force
	required to sustain such an elliptical path of Mars about the sun as a
	center of force located at a focal point of the ellipse.

	Building on Newton's description of the nature and universality of the
	gravitational force, scientists of the eighteenth century shifted
	their interest almost exclusively from direct to inverse problems.
	They used the combined gravitational forces of the sun and the other
	planets to predict and explain perturbations in the conic paths of
	planets and comets.  That interest continued through the nineteenth
	and twentieth centuries, and today scientists still concentrate upon
	the inverse problem rather than the direct one.

	In particular, in the last few decades there was a renewed interest in
	the Kepler problem as one of completely integrable Hamiltonian systems
	(IHS), the concept of which goes back to Liouville in 1897 \cite{lio}
	and Poincar\'{e} in 1899 \cite{po}.  Loosely speaking, IHS are
	dynamical systems admitting a Hamiltonian description, and possessing
	sufficiently many constants of motion.  Many of these systems are
	Hamiltonian systems with respect to two compatible symplectic
	structures \cite{mag1,gel,vil1,fil1} leading
	to a geometrical interpretation of the so-called recursion operator
	\cite{lax}.  The theory of integrable Hamiltonian systems, based on
	the use of the Nijenhuis torsion, is a part of the geometry of a
	particular class of manifolds, called Poisson-Nijenhuis manifolds
	\cite{mag2}.  In 1992, Marmo and Vilasi \cite{mar} constructed a
	recursion operator for the Kepler dynamics, and obtained related
	constants of motion.

	From the Magri works \cite{mag1,mag2}, it is known that the
	eigenvalues of the recursion operator of bi-Hamiltonian systems form a
	set of pairwise Poisson-commuting invariants \cite{bro4}. It is,
	however, worth noticing that two kinds of difficulties often arise,
	while investigating these systems: $(i)$ Firstly, it is in general
	very difficult to give locally an explicit second Hamiltonian
	structure for a given integrable Hamiltonian system \cite{rav} even if
	it is theoretically always possible in the neighborhood of a regular
	point of the Hamiltonian \cite{bro2}; $(ii)$ Secondly, the global or
	semi-local existence of such structures implies very strong conditions
	which are rarely satisfied \cite{bro,fer}.

	In 1996, R. Brouzet {\it et al.} defined a weaker notion under the
	name of {\it quasi-bi-Hamiltonian system} (QBHS) which relaxes these
	two difficulties for two degrees of freedom.  In 2000, G. Sparano {\it
		et al} constructed recursion operator for the Kepler dynamics, in
	the non-commutative case using the so-called Delauney action-angle
	coordinates \cite{spa}.  Further, in 2013, Hosokawa and Takeuchi
	\cite{kh} solved the same problem, but using the Runge-Lenz-Pauli
	vector, and got new constants of motion.  A bi-Hamiltonian formulation
	for a Kepler problem was also studied with Delaunay-type variables
	\cite{gri}.  In 2016, J. F. Cari\~{n}ena {\it et al.} \cite{car}
	investigated some properties of the Kepler problem related to the
	existence of quasi-bi-Hamiltonian structures.  In this work, we
	investigate the Kepler dynamics in a deformed phase space.

	The paper is organized as follows.  In Section \ref{sec_3}, we present
	the considered deformed phase space.  In Section \ref{sec_4}, we
	define, in action-angle coordinates, the deformed Hamiltonian
	function, symplectic form and vector field describing the Kepler
	dynamics.  In Section \ref{sec_6}, we construct recursion operators,
	and compute the associated integrals of motion. In Section
	\ref{sec_7}, we give an alternative Hamiltonian description for the
	dynamical systems and obtain associated recursion operators in a
	non resonant case. In Section \ref{sec_8}, we study the existence of
	quasi-bi-Hamiltonian structure for the considered Kepler dynamics. In
	Section \ref{sec_9}, we end with some concluding remarks.

	\section{Deformed phase space and Kepler Hamiltonian}\label{sec_3}

	Let $\mathbb{R}_{0}^{3} = \mathbb{R}^{3} \backslash \{0,0,0\}$ be the
	configuration manifold $\mathcal{Q},$ and
	$\mathcal{T}^{\ast}\mathcal{Q} = \mathcal{Q} \times \mathbb{R}^{3}$ be
	the cotangent bundle with the local coordinates $(q,p).$ The cotangent
	bundle $\mathcal{T}^{\ast}\mathcal{Q}$ has a natural symplectic
	structure $\omega$ which, in local coordinates, is given by

	\[ \omega = \sum_{i = 1}^{3} dq^{i} \wedge dp_{i}.\] Since $\omega$ is
	non-degenerate, it induces the map $\Lambda$:
	$\mathcal{T}^{\ast}\mathcal{Q} \longrightarrow \mathcal{T}\mathcal{Q}$
	defined by

	\[ \Lambda = \sum_{i = 1}^{3} \dfrac{\partial}{\partial q^{i}}\wedge
	\dfrac{\partial}{\partial p_{i}},\] where $\mathcal{T}\mathcal{Q}$
	is the tangent bundle. The map $\Lambda$ is called the bivector field
	\cite{vil2} and used to construct the Hamiltonian vector field $X_{f}$
	of a Hamiltonian function $f$ by the relation
	\begin{equation}\label{Kvec}
	X_{f} = \Lambda df.
	\end{equation}
	The phase space deformation is here understood by replacing the usual
	product with the $\gamma-$star product, (also known as the Moyal
	product law) between two arbitrary functions of position and momentum
	\cite{vak,ma,khos1} :
	\begin{equation} \label{Eq_3_1} (f\ast_{\gamma}g) (q,p) =
	f(q_{i},p_{i})  \exp{\bigg(\dfrac{1}{2}\gamma^{ab}\overleftarrow{\partial}_{a}\overrightarrow{\partial_{b}}\bigg)}
	g(q_{j},p_{j})\Bigg|_{(q_{i},p_{i})= (q_{j},p_{j})},
	\end{equation}
	where
	\begin{equation} \label{Eq_3_2} \gamma^{ab} =
	\begin{pmatrix}
	\varTheta^{ij} & \delta^{i}_{j}  \\
	- \delta^{i}_{j}  & 0 \\
	\end{pmatrix},
	\end{equation}
	$\varTheta$ is an antisymmetric $n \times n$ matrix inducing the
	deformation in the coordinates.  Without loss of generality, we
	restrict our study to the first two terms of the $\ast_{\gamma}$
	deformed Poisson bracket expansion to obtain:

	\begin{equation}
	\{ f,g\}_{\gamma} = \varTheta^{ij}\dfrac{\partial{f}}{\partial{q^{i}}}\dfrac{\partial{g}}{\partial{q^{j}}} + \bigg(\dfrac{\partial{f}}{\partial{q^{i}}}\dfrac{\partial{g}}{\partial{p_{i}}} - \dfrac{\partial{f}}{\partial{p_{i}}}\dfrac{\partial{g}}{\partial{q^{i}}}  \bigg),
	\end{equation}
	giving
	\begin{equation}\label{Eq_3_4}
	\{q^{i},q^{j}\}_{\gamma} = \varTheta^{ij}, \ \{q^{i},p_{j}\}_{\gamma} = \delta^{i}_{j}, \  \{p_{i},p_{j}\}_{\gamma} =0.
	\end{equation}
	The Kepler Hamiltonian in $\mathcal{T}^{\ast}\mathcal{Q}$ takes the
	form:
	\begin{equation}
	H = \dfrac{p_{i}p^{i}}{2m} + V(r),
	\end{equation}
	yielding the Hamilton's equations:
	\begin{equation} \label{eq2} \dot{q}^{i} := \{ q^{i} ,H\}_{\gamma}=
	\dfrac{p^{i}}{m} +
	\varTheta^{ij}\dfrac{\partial{V(r)}}{\partial{q^{j}}} ; \quad
	\dot{p}_{i} := \{ p_{i},H\}_{\gamma}= -
	\dfrac{\partial{V(r)}}{\partial{q^{i}}}
	\end{equation}
	and the following correction to the Newton second law \cite{ro1} :
	\begin{equation}
	m\ddot{q}^{i} = -\dfrac{q^{i}}{r}\dfrac{k}{r^{2}} + m\varepsilon^{ijk}\dot{q}^{j}\varOmega^{k} + m \varepsilon^{ijk}q^{j}\dot{\varOmega}^{k},
	\end{equation}
	where the deformation parameter
	$\varTheta^{ij} = \varepsilon^{ijk}\alpha^{k},$ and the angular
	velocity
	\[ \varOmega^{i} = \dfrac{k}{r^{3}}\alpha^{i}, i= 1,2,3. \] Setting
	the deformation parameter $\alpha^{i} = \delta^{i3}\alpha$ transforms
	$H$ into

	\begin{equation}\label{eq4'}
	H = \dfrac{m}{2}\bigg[ (\dot{q}^{1} - q^{2}\varOmega)^{2} + (\dot{q}^{2} + q^{1}\varOmega)^{2} + (\dot{q}^{3})^{2}\bigg] - \dfrac{k}{r},
	\end{equation}
	which is reduced to:
	\begin{equation}\label{Ham}
	H = \dfrac{p_{r}^{2}}{2m} + \dfrac{p_{\varphi_{\alpha}}^{2}}{2mr^{2}} - \dfrac{k}{r}
	\end{equation}
	in spherical coordinates $(r,\upsilon,\varphi)$, and equatorial orbit
	corresponding to $\upsilon = \dfrac{\pi}{2},$ where
	$ p_{r} = m\dot{r}$ and
	$ p_{\varphi_{\alpha}} = mr^{2} \dot{\varphi}_{\alpha},$ with
	$\dot{\varphi}_{\alpha} = (\dot{\varphi} + \varOmega)$ and
	$\varphi_{\alpha} = (\varphi + \varOmega t) \in (0, 2\pi ).$

	Equation \eqref{eq4'} encodes the information on the phase space deformation through  $ \varOmega,$ which depends on the deformation parameter $\alpha.$ However, it can  evidently be interpreted as equivalent
	to
	the Hamiltonian for a charged particle in a homogeneous, independent of
	time,  magnetic field along $z$ axis, and the central Newtonian gravitational
	field in the usual commutative space.

	Now considering the coordinate system
	$(r,\varphi_{\alpha},p_{r},p_{\varphi_{\alpha}}),$ and using
	\eqref{Kvec}, we get the following Hamiltonian vector field:
	\begin{equation}\label{Hamvk}
	X_{H} = \dfrac{1}{m}\Bigg[p_{r}\dfrac{\partial}{\partial r} - \dfrac{1}{r^{3}}\bigg( - p_{\varphi_{\alpha}}^{2}  + mkr \bigg)\dfrac{\partial}{\partial p_{r}}  + \dfrac{p_{\varphi_{\alpha}}}{m r^{2}}\dfrac{\partial}{\partial\varphi_{\alpha}}\Bigg].
	\end{equation}

	\section{Hamiltonian system in the action-angle
		coordinates} \label{sec_4}

	The Hamiltonian function \eqref{Ham} does not explicitly depend on the
	time.  Then, setting $V= W - Et, $ it is possible to find a complete
	integral for the equation of motion by using the method of variable
	separation:
	\begin{equation}
	W = W_{r}(r) + W_{\varphi_{\alpha}} (\varphi_{\alpha}).
	\end{equation}
	In this case, the Hamilton-Jacobi equation \cite{arn} is reduced to
	\begin{equation}
	E= \dfrac{1}{2m}\bigg( \dfrac{\partial{W}}{\partial{r}}\bigg)^{2} + \dfrac{1}{2mr^{2}}\bigg( \dfrac{\partial{W}}{\partial{\varphi_{\alpha}}}\bigg)^{2} - \dfrac{k}{r},
	\end{equation}
	leading to the following set of equations:
	\begin{equation*}
	\begin{cases}
	\bigg( \dfrac{dW_{\varphi_{\alpha}} (\varphi_{\alpha})}{d\varphi_{\alpha}}\bigg)^{2}= D_{\varphi_{\alpha}}^{2} \\
	-r^{2}\bigg(\dfrac{dW_{r}(r)}{dr}\bigg)^{2} + 2mr^{2}E + 2mrk =
	D_{\varphi_{\alpha}}^{2},
	\end{cases}
	\end{equation*}
	where $D_{\varphi_{\alpha}}$ is constant.  In the compact case
	\cite{vil2}, characterized by $E < 0,$ we can introduce the action
	variables \cite{abra} $J_{r}$ and $J_{\varphi_{\alpha}}$ such that:
	\begin{equation*}
	\begin{cases}
	J_{\varphi_{\alpha}}= \dfrac{1}{2\pi}\oint \dfrac{dW_{\varphi_{\alpha}}  (\varphi_{\alpha})}{d\varphi_{\alpha}}d\varphi_{\alpha} \\
	J_{r} =\dfrac{1}{2\pi} \oint \dfrac{dW_{r}(r)}{dr}dr.
	\end{cases}
	\end{equation*}
	Using the method of residue \cite{abl,vil2}, we get
	\begin{equation*}
	J_{r} = - p_{\varphi_{\alpha}} + \dfrac{mk}{\sqrt{-2mE}}, \ \ D_{\varphi_{\alpha}} = p_{\varphi_{\alpha}},
	\end{equation*}
	and the integrable system \cite{Bog}: {\small
		\begin{equation}\label{lio3}
		\begin{cases}
		\dot{J}_{i}= 0
		\\
		\dot{\varphi}^{i}= \dfrac{\partial H}{\partial J_{i}},
		\end{cases}
		\Rightarrow
		\begin{cases}{ll}
		J_{1}= J_{r}; \ J_{2}= J_{\varphi_{\alpha}}
		\\
		\varphi^{1}= \dfrac{mk^{2}}{(J_{1} + J_{2} )^{3}}t;\ \varphi^{2}
		= \dfrac{mk^{2}}{(J_{1} + J_{2} )^{3}}t, \ \varphi^{1}(0) =
		\varphi^{2} (0) = 0.
		\end{cases}
		\end{equation}}

	\begin{proposition}
		In action-angle coordinates $(J,\varphi),$ the Hamiltonian $H,$ the
		symplectic form $ \omega$ , and the Hamiltonian vector field $X_{H}$
		are, respectively:
		\begin{align} \label{Bil0} & H = E = - \dfrac{mk^{2}}{2(J_{1} +
			J_{2})^{2}}; \ \omega = \sum_{h = 1}^{2}dJ_{h}\wedge
		d\varphi^{h}, \\ \label{Bil2} \mbox{and} \cr X_{H} &= \{H,.\} :=
		\dfrac{mk^{2}}{(J_{1} + J_{2} )^{3}}\bigg(
		\dfrac{\partial}{\partial{\varphi^{1}}} +
		\dfrac{\partial}{\partial{\varphi^{2}}}\bigg),
		\end{align}
		where $\{. ,.\}$ is the usual Poisson bracket.
	\end{proposition}

	\section{Recursion operators } \label{sec_6} Let us define a $2$-form
	$\omega_{1}$ and a vector field $\Delta,$
	\begin{equation}
	\omega_{1}: = \sum_{h,k = 1}^{2} S_{h}^{k}dJ_{k}\wedge d\varphi^{h} = \sum_{h = 1}^{2}d\lambda_{h}\wedge d\varphi^{h} , \quad \Delta := \lambda_{h} \dfrac{\partial}{\partial{J_{h}}},
	\end{equation}
	where $S =
	\begin{pmatrix}
	J_{1} & J_{2}    \\
	J_{2} & J_{1}
	\end{pmatrix},\quad
	\begin{cases}
	\lambda_{1} = \dfrac{1}{2}\bigg( J_{1}^{2} + J_{2}^{2} \bigg) \\
	\lambda_{2} = J_{2}J_{1},
	\end{cases}$ such that $\omega_{1}$ is the Lie derivative of the
	symplectic form $\omega$ in \eqref{Bil0} with respect to the vector
	field $\Delta,$ $i.e.$:
	\[\mathcal{L}_{\varDelta}\omega = \omega_{1}. \]

	The vector field $\Delta$ generates a sequence of finitely many
	(Abelian) symmetries according to the following scheme:
	\[ X_{i + 1} := [X_{i}, \Delta]_{\mu}= \dfrac{2}{\mu}( X_{i}(\Delta) -
	\Delta(X_{i})),\] where $\mu = 3 - i, \ i=0,1,2 $ and
	$X_{0} = X_{H}$ in \eqref{Bil2}.  The $X_i$'s are given by
	\begin{align} \label{X_1} & X_{0} = \dfrac{mk^{2}}{(J_{1} + J_{2}
		)^{3}}\bigg( \dfrac{\partial}{\partial{\varphi^{1}}} +
	\dfrac{\partial}{\partial{\varphi^{2}}}\bigg), \quad X_{1} =
	\dfrac{mk^{2}}{(J_{1} + J_{2})^{2}}\bigg(
	\dfrac{\partial}{\partial{\varphi^{1}}} +
	\dfrac{\partial}{\partial{\varphi^{2}}}\bigg), \\ \label{X_2} &
	X_{2} = \dfrac{mk^{2}}{(J_{1} + J_{2} )}\bigg(
	\dfrac{\partial}{\partial{\varphi^{1}}} +
	\dfrac{\partial}{\partial{\varphi^{2}}}\bigg), \quad X_{3} =
	mk^{2}\bigg( \dfrac{\partial}{\partial{\varphi^{1}}} +
	\dfrac{\partial}{\partial{\varphi^{2}}}\bigg),
	\end{align}
	are:
	\begin{itemize}
		\item[(i)] in involution, \textit{i.e.},
		\begin{equation} [X_{h}, X_{k}]_{\mu} = 0, \quad h,k = 0,1,2,3, \ \
		\mu = 1,2,3.
		\end{equation}
		\item[(ii)] Hamiltonian vector fields, $i. e.$, can be expressed as:
		\begin{equation}
		X_{i} = \{H_{i}, .\} = \{H_{i + 1}, .\}_{_{1}}, \quad \ i= 0, 1, 2,
		\end{equation}
	\end{itemize}
	with respect to the Poisson bracket $ \{ .,.\}_{_{1}}$ defined by
	\begin{equation}
	\{ f,g\}_{_{1}} := \sum_{h,k = 1}^{n}(S^{-1})^{h}_{k} \bigg(\dfrac{\partial{f}}{\partial{J_{k}}}\dfrac{\partial{g}}{\partial{\varphi^{h}}} - \dfrac{\partial{f}}{\partial{\varphi^{h}}}\dfrac{\partial{g}}{\partial{J_{k}}}\bigg),
	\end{equation}
	where \[ S^{-1} = \begin{pmatrix}
	\dfrac{J_{1}}{(J_{1} - J_{2}) (J_{1} + J_{2} )}& \dfrac{- J_{2} }{(J_{1} - J_{2}) (J_{1} + J_{2})} \\
	\dfrac{- J_{2} }{(J_{1} - J_{2}) (J_{1} + J_{2} )} & \dfrac{ J_{1}
	}{(J_{1} - J_{2}) (J_{1} + J_{2})}
	\end{pmatrix},\] and
	\begin{equation}
	H_{0} = \dfrac{- mk^{2}}{2(J_{1} + J_{2} )^{2}}, \ H_{1} = \dfrac{- mk^{2}}{(J_{1} + J_{2} )}, \
	H_{2} =  mk^{2} \ln(J_{1} + J_{2} ), \ H_{3} = mk^{2}(J_{1} + J_{2} ).
	\end{equation}
	\begin{proposition}
		The recursion operator for the Kepler dynamics in the action-angle
		coordinates $(J,\varphi)$ is given by:
		\[
		T = \sum_{h,k}(S)_{k}^{h}\bigg(
		\dfrac{\partial}{\partial{J_{h}}}\otimes dJ_{k} +
		\dfrac{\partial}{\partial{\varphi_{h}}}\otimes d\varphi_{k}\bigg),
		\text{ where } \ S =
		\begin{pmatrix}
		J_{1} & J_{2}    \\
		J_{2} & J_{1}
		\end{pmatrix},\] $\mathcal{L}_{X_{l}}T = 0, \ (l = 0,1,2,3),$ and
		the Nijenhuis torsion vanishes, $i.e.,$ \\
		$(\mathcal{N}_{T})^{h}_{ij}:=T^{k}_{i}\dfrac{\partial{T^{h}_{j}}}{\partial{J^{k}}}
		- T^{k}_{j}\dfrac{\partial{T^{h}_{i}}}{\partial{J^{k}}} +
		T^{h}_{k}\dfrac{\partial{T^{k}_{i}}}{\partial{J^{j}}} -
		T^{h}_{k}\dfrac{\partial{T^{k}_{j}}}{\partial{J^{i}}}= 0, \ (i,j,k,
		h = 1,2).$
	\end{proposition}
	Consider the constants of motion \cite{ro2},
	\[H \ \text{in} \ \eqref{Bil0}, \ M = mr^{2} (\dot{\varphi} + 2
	\varOmega), \ \text{ and } \ L_{\alpha} = M + m\alpha H,\] $i.e.,$
	\begin{equation}
	\{H,M\}= 0, \ \{H,L_{\alpha}\}= 0 ,\ \{M,L_{\alpha}\} = 0.
	\end{equation}
	Then, there exist functions $\phi_{1}$, $\phi_{2}$ satisfying
	\begin{equation} \label{Bil1} \omega' = d\xi_{1}\wedge d\phi_{1} +
	d\xi_{2}\wedge d\phi_{2},
	\end{equation}
	such that the equations of motion in the coordinate system
	$(\xi, \phi)$ are
	\begin{equation}\label{fix1}
	\begin{cases}
	\dot{\xi}_{i}= 0
	\\
	\dot{\phi}^{i}= \dfrac{\partial H'}{\partial \xi_{i}},
	\end{cases}
	\Rightarrow
	\begin{cases}
	\xi_{i}= cst
	\\
	\phi^{i}= \dfrac{\partial H'}{\partial \xi_{i}}t , \quad
	\phi^{i}(0) = 0,\;\;i \in \{ 1,2 \},
	\end{cases}
	\end{equation}
	where $ \xi_{1} = L_{\alpha},$ $ \xi_{2} = M,$
	$H'= \dfrac{1}{m\alpha} (\xi_{1} - \xi_{2}).$ We get the relationships
	\begin{align}\label{fix}
	& J_{1} = - \xi_{2} +   \varpi + \sqrt{\dfrac{m^{2}\alpha k^{2}}{2(\xi_{2} -\xi_{1})}}; \quad J_{2} = \xi_{2} - \varpi , \\
	\label{fix3}
	&\varphi^{1}=  \dfrac{2\sqrt{2\alpha}}{m\alpha k}(\xi_{2} - \xi_{1} )^{3/2}\phi^{1}; \quad
	\varphi^{2}= -  \dfrac{2\sqrt{2\alpha}}{m\alpha k}(\xi_{2} - \xi_{1} )^{3/2}\phi^{2},
	\end{align}
	where $\varpi = mr^{2}\varOmega,$ $\xi_{2}> \xi_{1}>0, \alpha > 0 .$
	Finally, we arrive at:
	\begin{proposition}
		In the coordinate system $(\xi,\phi),$ the Hamiltonian function
		$H',$ the symplectic form $ \omega'$, the Hamiltonian vector field
		$X'_{H}$ and the recursion operator $T'$ are, respectively:
		\begin{align}
		H' &= \dfrac{1}{m\alpha} (\xi_{1} - \xi_{2}) ; \
		\omega'= \sum_{h = 1}^{2}d\xi_{h}\wedge d\phi^{h}; \
		X'_{H'} = \dfrac{1}{m\alpha} \bigg(\dfrac{\partial}{\partial{\phi_{1}}} - \dfrac{\partial}{\partial{\phi_{2}}}   \bigg) \\
		T' &= \sum_{i = 1}^{2} R_{i}\Bigg(\dfrac{\partial}{\partial{\xi_{i}}} \otimes d\xi_{i} + \dfrac{\partial}{\partial{\phi_{i}}} \otimes d\phi_{i}\Bigg), \text{ where} \ R =
		\begin{pmatrix}
		\xi_{1} & 0 \\
		0 & \xi_{2}
		\end{pmatrix}.
		\end{align}
	\end{proposition}
	Two interesting cases deserve investigation:
	\begin{itemize}
		\item[I)] Introduce the Laplace-Runge-Lenz (LRL) vector $A$ given by
		\cite{gol}
		\begin{equation}
		A = p \times L - mk\dfrac{q}{r},
		\end{equation}
		where $p$ is the momentum vector, $q$ is the position vector of the
		particle of mass $m$, and $L$ is the angle momentum vector,
		$L = q\times p$ \cite{t}. We obtain:
		\begin{equation} \label{agu} L_{1} = 0; \ L_{2} = 0; \ L_{3} =
		mr^{2} \dot{\varphi}_{\alpha} = p_{\varphi_{\alpha}}.
		\end{equation}
		\begin{equation}
		A_{1} = C \sin\beta  + D \cos\beta; \
		A_{2} = C \cos\beta - D\sin\beta; \
		A_{3} = 0,
		\end{equation}
		\begin{align}
		\{A_{1}, H\} &:= \Bigg(\dfrac{\partial{A_{1}}}{\partial{r}}\dfrac{\partial{H}}{\partial{p_{r}}} - \dfrac{\partial{A_{1}}}{\partial{p_{r}}}\dfrac{\partial{H}}{\partial{r}}\Bigg) + \Bigg(\dfrac{\partial{A_{1}}}{\partial{\varphi_{\alpha}}}\dfrac{\partial{H}}{\partial{p_{\varphi_{\alpha}}}} - \dfrac{\partial{A_{1}}}{\partial{p_{\varphi_{\alpha}}}}\dfrac{\partial{H}}{\partial \varphi_{\alpha}}\Bigg) \cr
		&\hphantom{:=\Bigg(\dfrac{\partial{A_{1}}}{\partial{r}}\dfrac{\partial{H}}{\partial{p_{r}}} - \dfrac{\partial{A_{1}}}{\partial{p_{r}}}\dfrac{\partial{H}}{\partial{r}}\Bigg)} \cr
		&= \dfrac{3 k \alpha p_{r}}{mr^{4}} (D \sin\beta - B\cos\beta), \\
		\{A_{2}, H\}  & = \dfrac{3 k \alpha p_{r}}{mr^{4}} (B \sin\beta - D\cos\beta),
		\end{align}
		where
		\[ C = -p_{r}p_{\varphi_{\alpha}}\cos\varphi_{\alpha} +
		\dfrac{p^{2}_{\varphi_{\alpha}}}{r} \sin\varphi_{\alpha} -
		mk\sin\varphi_{\alpha}\] and
		\[ D = p_{r}p_{\varphi_{\alpha}}\sin \varphi_{\alpha} +
		\dfrac{p^{2}_{\varphi_{\alpha}}}{r} \cos\varphi_{\alpha} -
		mk\cos\varphi_{\alpha}, \text{ and } \ \beta =\varOmega t.\]
		\begin{remark}
			We have:
			\begin{itemize}
				\item[(i)] The $A_{i}, \;\; i= 1, 2, 3,$ commute with the
				Hamiltonian $H$ in \eqref{Ham}, $i.e.,$ $\{A_{i},H\} = 0,$ if
				\begin{equation}
				\beta = \dfrac{\pi}{4}; \quad \dfrac{p_{r}p_{\varphi_{\alpha}}}{ \dfrac{p^{2}_{\varphi_{\alpha}}}{r} - mk}= -\cot(\beta + \varphi_{\alpha}),\;\;(\beta + \varphi_{\alpha}) \in (0, \pi).
				\end{equation}
				\item[(ii)]
				$ \{A_{1}, A_{2}\} = ( - 2 mH + \dfrac{3 k \alpha
					p_{r}}{r^{4}})p_{\varphi_{\alpha}}, \ \{A_{1}, L_{3}\} =
				A_{2},$ and $\{A_{2}, L_{3}\} = A_{1}.$
				\item[(iii)] Setting $L_{3} =A_{3}$ and
				$ p^{2}_{\varphi_{\alpha}} = r[2mk -r + p_{r} (3\varOmega -
				rp_{r})] \equiv A^{2}_{3},$ then, the
				$A_{i} {\textquoteright s}$ generate an $su(2)$ Lie algebra,
				$i.e.,$ $\{A_{i}, A_{j}\} = \varepsilon_{ijl}A_{l}.$
			\end{itemize}
		\end{remark}
		\item[II)] Consider a scaled Runge-Lenz-Pauli vector $\Gamma,$ defined
		on the domain
		$\{(q,p)\in \mathcal{T^{\ast}}(\mathbb{R}^{3}\backslash \{0,0,0\})|
		H(q,p) < 0\}$ by
		\begin{equation} \label{rlp} \Gamma= \dfrac{1}{\sqrt{-2mH}}A,
		\end{equation}
		where $H$ is the Hamiltonian function given in \eqref{Ham}. The
		components $\Gamma_{i}$ are:
		\begin{equation}
		\Gamma_{1} =\dfrac{1}{\sqrt{-2mH}}( C \sin\beta  + D \cos\beta); \
		\Gamma_{2} = \dfrac{1}{\sqrt{-2mH}}( C \cos\beta - D\sin\beta); \
		\Gamma_{3} = 0,
		\end{equation}
		with
		\begin{equation}
		|\Gamma|^{2} = - \dfrac{mk^{2}}{2 H} + L^{2}_{3}.
		\end{equation}
		The quantities $H$, $|\Gamma|^{2},$ and $L_{3}$ are in involution,
		$i.e.,$
		\[
		\{|\Gamma|^{2},L_{3}\} = 0 ,\ \ \{|\Gamma|^{2},H\} = 0, \ \
		\{L_{3},H\} = 0.
		\]
	\end{itemize}
	Putting $\pi_{1} = |\Gamma|^{2}$ and
	$\pi_{2} = \ p_{\varphi_{\alpha}},$ the equations of motion in the
	$(\pi, \chi)$ system become:
	\begin{equation}\label{fix4}
	\begin{cases}
	\dot{\pi}_{i}= 0
	\\
	\dot{\chi}^{i}= \dfrac{\partial H''}{\partial \pi_{i}},\;\;H'' =
	\dfrac{m k^{2}}{2(\pi_{2}^{2} -\pi_{1})}
	\end{cases}
	\Rightarrow
	\begin{cases}
	\pi_{i}= cst, \;\; i= 1, 2.
	\\
	\chi^{i}= \dfrac{\partial H''}{\partial \xi_{i}}t +
	\chi^{i}(0),\quad \chi^{i}(0)= 0.
	\end{cases}
	\end{equation}
	The relationships between $(J,\varphi)$ and $(\pi, \chi)$ are deduced
	as:
	\begin{equation}\label{PI1}
	J_{1} = - \pi_{1} +    \sqrt{\pi_{1} - \pi_{2}^{2}}; \ J_{2} = \pi_{2}; \ \chi^{1} =  \dfrac{1}{(J_{1} + J_{2} )}\varphi^{1}; \ \chi^{2} = - \dfrac{J_{2} }{(J_{1} + J_{2} )} \varphi^{2}.
	\end{equation}
	Finally, we get:
	\begin{proposition}
		In the coordinate system $(\pi,\chi),$ the Hamiltonian function
		$H'',$ the symplectic form $ \omega''$, the Hamiltonian vector field
		$X''_{H''}$ and the recursion operator $T''$ are given as follows:
		\begin{align}
		& H'' = \dfrac{m k^{2}}{2(\pi_{2}^{2} -\pi_{1})}; \
		\omega'' = \sum_{h = 1}^{2}d\pi_{h}\wedge d\chi^{h}; \
		X''_{H''} = \dfrac{mk^{2}}{2(\pi_{2}^{2} - \pi_{1})^{2}} \bigg(\dfrac{\partial}{\partial{\chi^{1}}} - 2\pi_{2} \dfrac{\partial}{\partial{\chi^{2}}}\bigg) \\
		& T'' =  \sum_{i = 1}^{2} F_{i}\Bigg(\dfrac{\partial}{\partial{\pi_{i}}} \otimes d\pi_{i} + \dfrac{\partial}{\partial{\chi^{i}}} \otimes d\chi^{i}\Bigg), \text{ where } \  F =
		\begin{pmatrix}
		\pi_{1} & 0 \\
		0 & \pi_{2}
		\end{pmatrix}.
		\end{align}
	\end{proposition}
	\section{Alternative Hamiltonian description} \label{sec_7}

	Let
	\begin{equation} \label{AD} \Upsilon = J_{1}X_{1} + J_{2}X_{2}
	\end{equation}
	be a dynamical system on the manifold $\mathcal{T}^{\ast}\mathcal{Q}$,
	with $ X_{1}$ and $ X_{2}$ obtained in \eqref{X_1} and
	\eqref{X_2}. The relation \eqref{AD} can be rewritten as:
	\begin{equation} \label{AD1} \Upsilon = \nu_{a}X^{a} +\nu_{e}X^{e},
	\end{equation}
	where
	\begin{align} \label{nu}
	&\nu_{a} = -2 H_{a}, \ \nu_{e} = H_{e}; \ H_{a} = J_{1} H_{0}, \  H_{e} = J_{2} H_{1}; \ X^{a} = \dfrac{\partial}{\partial\Phi^{a}},\ X^{e}= \dfrac{\partial}{\partial\Phi^{e}}, \\
	& \dfrac{\partial}{\partial\Phi^{a}} = \bigg(
	\dfrac{\partial}{\partial\varphi^{1}} +
	\dfrac{\partial}{\partial\varphi^{2}} \bigg), \quad
	\dfrac{\partial}{\partial\Phi^{e}} = - \bigg(
	\dfrac{\partial}{\partial\varphi^{1}} +
	\dfrac{\partial}{\partial\varphi^{2}} \bigg).
	\end{align}
	The vector fields $X^{a}, X^{e}$ and the $C^{\infty}-$functions
	$ H_{a}, H_{e}$ satisfy the following properties:
	\begin{equation} [X^{i}, X^{j}] = 0; \quad \mathcal{L}_{X^{i}} H_{i} =
	0, \ \ i,j \in \{ a,e\}.
	\end{equation}
	Let $\mathcal{ N}$ be an open dense submanifold of
	$\mathcal{T}^{\ast}\mathcal{Q}$ on which $\Upsilon$ is explicitly
	integrable such that:
	\begin{equation} \label{c1} X^{a}\wedge X^{e}\neq 0; \quad dH_{a}
	\wedge dH_{e} \neq 0.
	\end{equation}
	Now, considering the coordinate system $(H,\Phi)$ with $\Phi^{i},$
	$i \in \{ a,e\},$ which are closed differential 1-forms, the equations
	of motion of $\Upsilon$ are given by

	\begin{equation}
	\dot{\Phi}^{a}= -2H_{a}; \
	\dot{\Phi}^{e} = H_{e}; \
	\dot{H}_{a} = 0; \
	\dot{H}_{e} = 0,
	\end{equation}
	with the functions $H_{a}$ and $H_{e}$ obeying the condition
	\eqref{c1}. We can construct a closed $2$-form, for $i \in \{a,e\}$,
	\begin{equation} \label{om} \tilde{\omega} = \sum_{i}
	df^{i}(H_{i})\wedge d\Phi^{i},
	\end{equation}
	which is non degenerate as long as $df^{a} \wedge df^{e}\neq 0 $, and
	\begin{equation} \label{c2} \iota_{_{X^{i}}}\tilde{\omega} = -df^{i};
	\quad \iota_{_{\Upsilon}}\tilde{\omega} = - \sum_{i} \nu_{i} df^{i};
	\quad \sum_{i} d\nu_{i}\wedge df^{i} = 0.
	\end{equation}
	Notice that \eqref{c2} is a necessary condition for
	$\iota_{_{\Upsilon}}\tilde{\omega} $ to be exact, $i.e.,$ it is
	closed.  Since $d\nu_{a} \wedge d\nu_{e} \neq 0$, the solutions of
	\eqref{c2} are given by linear functions \cite{gl}
	\begin{equation}
	f^{i} = \sum_{j} L^{ij}\nu_{j}, \quad i, j \in \{a,e\}, \text{ where } \  L=       \begin{pmatrix}
	-1/2 & 0 \\
	0 & 1
	\end{pmatrix}.
	\end{equation}
	Then, we get :
	\begin{equation} \label{fi} f^{a}= - \dfrac{1}{2}\nu_{a}; \quad f^{e}=
	\nu_{e}.
	\end{equation}
	From \eqref{nu} and \eqref{fi}, we can rewrite \eqref{om} in the new
	coordinate system $(\nu, \Phi)$ as:

	\begin{equation} \label{om1} \tilde{\omega} = \sum_{i}
	df^{i}(\nu_{i})\wedge d\Phi^{i}, \ \ i \in \{a,e\},
	\end{equation}
	leading to the following form:
	\begin{equation} \label{om2} \tilde{\omega} = -
	\dfrac{1}{2}d\nu_{a}\wedge d\Phi^{a} + d\nu_{e}\wedge d\Phi^{e}.
	\end{equation}
	The corresponding Hamiltonian description for $\Upsilon$ is given with
	the following quadratic Hamiltonian function
	\begin{equation} \label{Ha2} \tilde{ H } = - \dfrac{1}{4}\nu_{a}^{2} +
	\dfrac{1}{2}\nu_{e}^{2}.
	\end{equation}
	In addition, from \cite{vil2} other symplectic structures of the form
	\eqref{om1} can be constructed, in which any $f_{i} $ depending only
	on the corresponding frequency $\nu_{i}, \ i \in \{a,e\}$, will be
	admissible as long as $\tilde{\omega}_{b}, \ b \in \{1,...,n\}, $ is
	non-degenerate, $i.e.$, as long as $df^{a} \wedge df^{e} \neq 0.$ From
	above, putting:
	\begin{equation} \label{fi1} f^{a}= \nu_{a}; \ f^{e}= \nu_{e} \ \text{
		and } f^{a}= \nu_{a}^{2}; \ f^{e}= \nu_{e}^{2}
	\end{equation}
	we obtain, respectively:
	\begin{equation} \label{om3} \tilde{\omega}_{1} = d\nu_{a}\wedge
	d\Phi^{a} + d\nu_{e}\wedge d\Phi^{e} \ \text{ and } \
	\tilde{\omega}_{2} = 2\nu_{a}d\nu_{a}\wedge d\Phi^{a} +
	2\nu_{e}d\nu_{e}\wedge d\Phi^{e}.
	\end{equation}
	Then, the $(1,1)-$tensor field
	$\mathcal{ T} = \tilde{\omega}_{2} \circ \tilde{\omega}_{1}^{-1}$ is
	constructed, taking the form
	\begin{equation}
	\mathcal{ T} = \mathcal{ T}_{1} + \mathcal{ T}_{2},
	\end{equation}
	where
	\begin{equation}
	\mathcal{ T}_{1} =2 \nu_{a} \bigg(\dfrac{\partial}{\partial\nu_{a}}\otimes d\nu_{a} + \dfrac{\partial}{\partial \Phi^{a}}\otimes d\Phi^{a} \bigg) \
	\text{ and } \
	\mathcal{ T}_{2} = 2 \nu_{e} \bigg(\dfrac{\partial}{\partial\nu_{e}}\otimes d\nu_{e} + \dfrac{\partial}{\partial \Phi^{e}}\otimes d\Phi^{e} \bigg).
	\end{equation}
	Finally, basing on \cite{t,ta,ta2}, $\mathcal{ T}_{1}$
	and $\mathcal{ T}_{2}$ are recursion operators for the dynamical
	system $\Upsilon$. Hence, $\mathcal{ T}$ is also a recursion operator
	for the dynamical system $\Upsilon$ as a sum of two recursion
	operators.

	\section{Quasi-bi-Hamiltonian structures} \label{sec_8} Basing on
	\cite{bro} and \cite{car}, in this part, we investigate the recursion
	operators for quasi-bi-Hamiltonian structures.
	\begin{definition}
		A Hamiltonian vector field $Y$ on a symplectic manifold
		$(\mathcal{M}, \omega) $ is called quasi-bi-Hamiltonian if there
		exist another symplectic structure $\omega_{1}$, and a
		nowhere-vanishing function $g$, such that $g Y$ is a Hamiltonian
		vector field with respect to $ \omega_{1},$ $i.e.,$
		\begin{equation}
		\iota_{_{Y}}\omega_{0} = -dH_{0}; \quad
		\iota_{_{gY}}\omega_{1} =\iota_{_{Y}}(g\omega_{1})  = -dH_{1},
		\end{equation}
		where $H_{0}$ and $H_{1}$ are integrals of motion for the
		Hamiltonian vector field $Y.$ $g\omega_{1}$ is not closed in
		general.
	\end{definition}
	A consequence of this definition is that the pair
	$(\omega_{0}, \omega_{1})$ determines a $(1, 1)$-tensor field $T$
	defined as $T := \hat{\omega}_{0}^{-1}\circ \hat{\omega}_{1},$ that
	is, $\omega_{0}(Y, X) = \omega_{1}(T Y, X ),$ where $X, \ Y$ are two
	Hamiltonian vector fields, and $\hat{\omega} := \iota_{_{Y}}\omega.$
	In the action-angle coordinates $(J,\varphi),$ the decomposition of
	the symplectic form

	$ \omega' = \omega'_{1} + \omega'_{2}$,
	\begin{align}
	\omega'_{1} &= dJ_{1}\wedge d\varphi^{1} - \bigg( \dfrac{2 (J_{1} + J_{2})^{3}}{m^{2} k^{2} \alpha} + 1   \bigg) dJ_{2}\wedge d\varphi^{2}  \\
	\omega'_{2} &=    - dJ_{1}\wedge d\varphi^{2} + \bigg( \dfrac{2 (J_{1} + J_{2})^{3}}{m^{2} k^{2} \alpha} + 1   \bigg) dJ_{2}\wedge d\varphi^{1},
	\end{align}
	shows that:
	\begin{itemize}
		\item[(i)] $\omega'_{1}$ and $\omega'_{2}$ are not closed, $i.e.$,
		$d\omega'_{1} \neq 0, d\omega'_{2} \neq 0$, where $d$ is the
		exterior derivative. So, $\omega'_{1}$ and $\omega'_{2}$ are not
		symplectic.
		\item[(ii)]
		$ \iota_{_{X_{H}}}\omega'_{1} = -dh'_{1}, \
		\iota_{_{X_{H}}}\omega'_{2} = -dh'_{2}, \ \text{ where } \ h'_{1} =
		-h'_{2}= - \dfrac{2 J_{2}}{m\alpha}.  $
		\item[(iii)] The functions $ h'_{1}$ and $ h'_{2}$ are first integrals
		of $X_{H},$ $i.e., \ X_{H}( h'_{1}) = X_{H}( h'_{2}) = 0.$
	\end{itemize}

	\begin{proposition}
		The Hamiltonian vector field $X_{H}$ is quasi-bi-Hamiltonian with
		respect to the two $2$-forms $(\omega, \omega'_{1}).$ Idem for
		$(\omega, \omega'_{2}).$ The weaker $\omega'_{i}$ recursion operators
		are given by:
		\begin{align}
		\widetilde{ T}'_{1} & :=  \omega^{-1} \circ \omega'_{1}  \cr
		& = \dfrac{\partial}{\partial J_{1}}\otimes dJ_{1} + \dfrac{\partial}{\partial \varphi^{1}}\otimes d\varphi^{1} - (2K + 1) \bigg( \dfrac{\partial}{\partial J_{2}}\otimes dJ_{2}
		+  \dfrac{\partial}{\partial \varphi^{2}}\otimes d\varphi^{2}\bigg),
		\end{align}
		and
		\begin{align}
		\widetilde{T}'_{2} & := \omega^{-1} \circ \omega'_{2}  \cr
		& =  (2K + 1) \bigg(  \dfrac{\partial}{\partial J_{1}}\otimes dJ_{2} + \dfrac{\partial}{\partial \varphi^{2}}\otimes d\varphi^{1} \bigg)- \dfrac{\partial}{\partial \varphi^{1}}\otimes d\varphi^{2} - \dfrac{\partial}{\partial J_{2}}\otimes dJ_{1},
		\end{align}
		where \\
		$K =\dfrac{ (J_{1} + J_{2})^{3} }{m^{2}k^{2}\alpha}$ and the
		$2$-vector field
		$ \omega^{-1} = \dfrac{\partial}{\partial J_{i}}\wedge
		\dfrac{\partial}{\partial \varphi^{i} }.$
	\end{proposition}
	Similarly, $\omega''$ can be re-expressed as the sum of two $2$-forms
	as follows:
	\begin{equation}
	\omega'' =  \omega''_{1} +  \omega''_{2},
	\end{equation}
	where
	\begin{align}
	\omega''_{1} &= 2 dJ_{1}\wedge d\varphi^{1} - \bigg( 2J_{2} +  \dfrac{J_{2}(2J_{2} + 1)}{(J_{1} + J_{2})^{2}} \bigg)dJ_{2}\wedge d\varphi^{2} \\
	\omega''_{2} &=- J_{2}dJ_{1}\wedge d\varphi^{2}  - 2\dfrac{(1 + J_{1})\varphi^{1}}{(J_{1} + J_{2})} dJ_{1}\wedge dJ_{2}   + \bigg( 2  + \dfrac{2J_{2} + 1}{(J_{1} + J_{2})}\bigg)dJ_{2}\wedge d\varphi^{1} \cr
	& + \bigg( \dfrac{2}{(J_{1} + J_{2})} + \dfrac{2J_{2} + 1}{(J_{1} + J_{2})^{2}}\bigg)(J_{2} - 1)\varphi^{1}dJ_{2}\wedge dJ_{1} .
	\end{align}
	As above:

	\begin{itemize}
		\item[(iv)] $\omega''_{1}$ and $\omega''_{2}$ are not symplectic,
		$i.e.,$ $d\omega''_{1} \neq 0, d\omega''_{2} \neq 0$.
		\item[(v)]
		$ \iota_{_{X_{H}}}\omega''_{1} = -dh''_{1}, \quad
		\iota_{_{X_{H}}}\omega''_{2} = -dh''_{2}, $ where
		\begin{align}
		h''_{1} &= \dfrac{k^{2}m (3 J_{2}(8J_{2} - 6J_{1} + 3) + J_{1}(2J_{1} + 5))}{6(J_{1} + J_{2})^{3}}, \\
		h''_{2} &= \dfrac{ k^{2}m(J_{2}(-J_{2} - 3J_{1} + 12) + 8 J_{1})}{6(J_{1} + J_{2})^{3}}.
		\end{align}
		\item[(vi)] $h''_{1}$ and $h''_{2}$ are also first integrals of
		$X_{H},$ $i.e., X_{H}( h''_{1}) = X_{H}( h''_{2}) = 0.$
	\end{itemize}
	\begin{proposition}
		The Hamiltonian vector field $X_{H}$ is quasi-bi-Hamiltonian with
		respect to the two $2$-forms $(\omega, \omega''_{1}).$ Idem for
		$(\omega, \omega''_{2}).$ The weaker $\omega''_{i}$ recursion
		operators $\widetilde{T}''_{1}$ and $\widetilde{T}''_{2}$ are:
		\begin{align}
		\widetilde{T}''_{1} & := \omega^{-1} \circ \omega''_{1}  \cr
		& = 2\bigg(  \dfrac{\partial}{\partial \varphi^{1}}\otimes d\varphi^{1} + \dfrac{\partial}{\partial J_{1}}\otimes dJ_{1}\bigg) - J_{2}\bigg(2 + \dfrac{\widetilde{V}}{V^{2}} \bigg) \bigg( \dfrac{\partial}{\partial J_{2}}\otimes dJ_{2} +
		\dfrac{\partial}{\partial \varphi^{2}}\otimes d\varphi^{2}\bigg), \cr
		\\
		\widetilde{T}''_{2} & := \omega^{-1} \circ \omega''_{2}  \cr
		& = \bigg( 2 +  \dfrac{\widetilde{V}}{V}\bigg) \bigg(  \dfrac{\partial}{\partial J_{1}}\otimes dJ_{2} + \dfrac{\partial}{\partial \varphi^{2}}\otimes d\varphi^{1} \bigg) -
		J_{2} \bigg(  \dfrac{\partial}{\partial \varphi^{1}}\otimes d\varphi^{2} + J_{2}\dfrac{\partial}{\partial J_{2}}\otimes dJ_{1} \bigg)\cr
		& -\bigg(2 + \dfrac{\widetilde{V}}{V^{2}} (J_{2} - 1) \bigg)\varphi^{1} \bigg( \dfrac{\partial}{\partial \varphi^{1}}\otimes dJ_{2} - \dfrac{\partial}{\partial \varphi^{2}}\otimes dJ_{1}\bigg),
		\end{align}
		where $\widetilde{V} = 2 J_{2} +1, \ V = J_{1} + J_{2}.$
	\end{proposition}

	\section{Concluding remarks} \label{sec_9} In this paper, we have
	constructed recursion operators for the Kepler dynamics in a deformed
	phase space by considering the equatorial orbit, computed the
	associated integrals of motion, and proved the existence of
	quasi-bi-Hamiltonian structures for the Kepler dynamics.

	\subsection*{Acknowledgments}
	This work is supported by TWAS Research Grant RGA No.17-542 RG/
	MATHS/AF/AC\_G-FR3240300147.  The ICMPA-UNESCO Chair is in partnership
	with the Association pour la Promotion Scientifique de l'Afrique
	(APSA), France, and Daniel Iagolnitzer Foundation (DIF), France,
	supporting the development of mathematical physics in Africa.

redpajama_set_name: RedPajamaArXiv


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If someone told you last August that Leicester City, the club that narrowly avoided relegation in 2014/15, was to win the Premier League this season – you would have laughed at them in the face (and probably avoided them in the street next time you saw them). Their start of season odds to win the league were after all, 5000 to 1. But that is the beauty of football, you're only as good as your next 90 mins – reputations mean nothing. Leicester City, incredibly, are the Premier League champions. So how did this happen? What were some of the factors and stats at play with this miraculous and momentous triumph?
What Made Leicester City Premier League Champions?
The attacking duo of Riyad Mahrez and Jamie Vardy were ruthless in-front of goal all season, as a pair scoring 41 league goals for the season. To put that in perspective, six EPL teams failed to score more goals as a whole team, than both Riyad Mahrez and Jamie Vardy. What is even more incredible is that the pair only managed 9 league goals between them in Season 2014/15. The pair were t he two highest points scorers in the Fantasy Premier League this year, and with both priced below £7.5m, I can't recall a cheaper top scorer in recent Fantasy Premier League. Keeping these two goal machines is essential for Leicester to continue to press for European honors next season. Leicester's attack was the third best in the league, scoring 68 league goals at an average of 1.79 goals per game.
While Riyad Mahrez, Jamie Vardy and co. were banging the goals in at one end, at the other end Leicester defense was evolving into a force as the season progressed. With Daniel Drinkwater and N'Golo Kanté tirelessly providing cover from midfield, Leicester managed to keep 11 clean sheets from the second half of the season (19 matches). This was a massive improvement on the 4 kept in the first half of the season. This maturing showed that they could play both open attacking football, as well as conservative with a deadly counter attack. In 2014/15 Leicester managed 10 clean sheets for the season, but in their season as champions they had the fourth best defense in the league finishing with 15 clean sheets.
While Leicester City throughly deserve their EPL champions crown, the declining power of the traditional 'big clubs' as they transition, was surely a factor here. Chelsea massively underperformed with their player in-fighting leading to the sacking of Jose Mourinho. Louis van Gaal was still trying to bring life back into Manchester United. Liverpool parted ways with their manager too, and Manchester City failed to play the type of attacking football we've come to expect from them. This was a big missed opportunity for Arsenal, who will feel this was their year to deliver, given the quality of their squad. With Tottenham & Leicester both in the Champions League next season, the dynamic of the Premier League is changing – gone is the strangle-hold on the top places.
It's not completely known how much of Leicester's success this season was down to their manager Claudio Ranieri. Many critics raised an eyebrow when he was appointed manager last year, but Claudio Ranieri is a high performing manager with a cv of massive European clubs. Don't forget his last time managing in Premier League was a second place finish with Chelsea. His rotation policy during his time at Chelsea earned him the nickname "Tinkerman", at Leicester this season he settled on his preferred lineup early and only mage changes when necessary. Ranieri worked hard with the press to keep expectations off his players, keeping things lite to reduce the pressure. This calming influence was crucial as the season progressed and Leicester's bubble failed to break.
Was luck on Leicester's side this season? Sure, definitely. But isn't it with any team that wins the EPL? Certainly Chelsea has their fair share as Champions in season 2014/15. Leicester won 7 league matched by a scoreline of 1-0, and a further 7 more league wins by a margin of 1 goal. That's 14 matches that could have ended in draw's has things played out slightly different, 42 points could well be just 14 points or fewer next season. Importantly Leicester finished 10 points clear at the top of the table, which means they are well clear of some one-off fluke result.
Well played Leicester City, champions of the Premier League for 2015/16.


We've got a wide variety of participants coming to the LOD-LAM Summit, so suggesting a reading list is kind of tough. Keep in mind that participants range from technology staff, policy makers, developers, librarians, digital humanists, hackers and everywhere in between. I'm going to throw out some of my favorite books and articles, but please add more in the comments as this is by no means exhaustive. And if a lot of these names look familiar, it's because you've seen them on the participant list for the Summit.
Open Bibliographic Data Guide. This guide from JISC focuses more on open rather than linked data, but it's a critical first step toward Linked Open Data.
Programming the Semantic Web, Toby Segaran, Colin Evans, & Jamie Taylor. 2009. Great primer on graphs and plenty of example code.
Linked Data: Evolving the Web into a Global Data Space, Tom Heath and Christian Bizer. 2011. This is a great book, recently released, that provides a concise and in depth exploration into Linked Data, from conceptual overview to recipes for publishing data.
Rights and Licensing from JISC Open Bibliographic Data Guide. Recommendations for publishing Open Data for Libraries.
Digital Cultural Collections in an Age of Reuse and Remixing, Kristin R. Eschenfelder and Michelle Caswell. Nov. 2010. This study examines the various views and considerations of cultural institutions in allowing reuse of digital cultural works. It's based on a 2008 survey that is, in my opinion, just at the turning point of a rather radical cultural shift in opening metadata for reuse and sharing.
Recommendations for independent scholarly publication of data sets, Jonathan Rees. March 2010. These recommendations come from the perspective of the sciences but can equally be applied to the humanities, and embodies the shift toward sharing data for future use and portability.
I'm sure there are more articles to add to this list and please feel free to do so in the comments.
Smithsonian Information, Smithsonian linked open data?
We're continuing our discussions here at the Smithsonian about what we can do to make our data accessible in a linked, open manner. The other day I had some discussions with the staff at the Smithsonian American Art Museum (SAAM) about some possible projects that we can do that will have a short gestation time.
Looking forward to sharing more at the summit.
Just for the fun of it, I submitted a video postcard for Ignite Smithsonian, taking place 4/11/11. I was trying to capture the excitement of the emerging LODLAM community, in a 30 second spot.
and led the internet revolution?
Institutional decision makers can gain a firm grasp on the techniques, technology and terminology of Linked Open Data.
Collaborative strategy will be developed to forward the adoption and promotion of Linked Open Data in the humanities in the United States.
Actionable steps and strategies for sharing, publishing and licensing structured data on the Web will be widely distributed and publicized.
The NEH joins the Alfred P. Sloan Foundation in their generous support of the Summit. The Sloan Foundation will support international participants and those from the sciences and humanities. Sloan funding also enables us to continue outreach and forward momentum on LOD-LAM efforts during the year following the Summit, including meetings and panels at various industry and sector conferences globally.
The International Linked Open Data in Archives, Libraries, and Museums is the first gathering of its kind, and is being hosted by the Internet Archive in San Francisco, CA June 2-3, 2011.


Leinster will provide five teams for the All-Ireland schools competition this year which will be hosted by Villiers in Limerick on October 24-26.
Perennial title hopefuls St Andrews and Wesley both gained automatic qualification for the tournament while the other two places will come from a pair of playoff matches – Sutton Park will play King's Hospital while last year's cup finalists High School will play Kilkenny.
The two losing sides will subsequently play off against each other to decide the final place with all three fixtures set for September 17 in Grange Road, to be played back-to-back.
The tournament itself will see four Munster sides involved and seven from Ulster.
On the domestic scene, Leinster will hold a development league and cup this year in an effort to increase the base of schools playing and to bring some of those who have fallen back in recent years back into the fold.
Last year, Skerries and St Mark's both fielded teams after a few years absence in a tournament of this kind, won by Drogheda Grammar in which Sandford Park also took part.
Elsewhere, there will be Junior A tournaments this year to try to get the up-and-coming schools like Mount Temple, St Columba's, High School, Dundalk and Kilkenny back to A league hockey.
Last season, Kilkenny managed to reach the junior cup final without having played a league match and pushed a star-studded Wesley side all the way and it is hoped this can act as a catalyst.
In another new initiative, the Leinster schools committee, headed by Ronan Walsh, is currently working with Lucas Piccioli to appoint temporary coaches for struggling schools.


Dizziness is a feeling of imbalance that suddenly appears. Dizziness during pregnancy is common, having several origins.
In this article you can know the reasons that lead to dizziness in pregnancy, as well as some care you can have to avoid this sensation.
Dizziness in pregnancy are normal and natural symptoms at this stage, and there are several causes.
The two most common sources for feeling dizzy are changes in blood pressure or as a symptom of hypoglycemia.
In the first case, because of the increase in blood volume that occurs with the onset of gestation (on average, the circulatory system is 1.5 liters more than normal), the heart is still adapting to this higher volume, causing a drop in blood pressure.
In the second case, in case of inadequate feeding, the pregnant woman may have low sugar levels in the brain and blood, which will cause hypoglycaemia. One of its manifestations is dizziness.
If you often feel dizzy, then there are some precautions you can take to keep you from feeling that sensation again. Here are some precautions that will help you.
– When you wake up, you should not get up straight away. Instead, sit down and let yourself be like this for a few minutes.
– When sitting or lying down, stand up quietly. Sudden changes in position may cause dizziness.
– When you are sleeping, stand on your side, with your legs on a cushion. This position allows you to prevent the uterus from pressing on the vena cava.
– Eat often, without spending too much time between meals. That way, you will always keep your blood sugar and brain levels at a normal level.
When dizziness occurs, there may be a risk of falling. After all, dizziness is a feeling of imbalance. Then we leave some care that you should have when feeling dizzy, in order to reduce the risk of falling.
– When you feel dizzy, sit with your head between your legs or lie on your left side.
– Drink water or juice.
– If it is in a very hot and stuffy place, look for a cooler and cooler place.
Usually dizziness is fleeting, having nothing to worry about. However, if dizziness is very common, and if it is associated with palpitations, headache, or blurred vision, talk to your doctor as there may be other causes behind.
Is it safe to use metformin during pregnancy?
Antiprotozoal Drugs. Antiprotozoal Drugs MALARIA.


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Depending on your subscription we offer different maps for tour planning. The differences in the map data are described in detail in this article.
On the lower right corner of the map you can activate different overlays via "Maps and trails". Besides the OpenStreetMap you will find the maps "Outdooractive", "Topo" and "Alpine Club map". Under Style you can choose between Summer, Winter and Satellite.
To keep the paths better apart, you can activate the hiking, cycling, mountainbiking or horse riding network and the winter sports overlay via the ways and activities. Slope angle, notices & closures, and avalanche location are available under Additional Layers.


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Had a busy day? Need to wash your hair? Can't find your trainers or have have a headache? If you want to get into shape and improve your health for summer then you need to put aside those excuses and start getting motivated.
This is the oldest line in the book. If we are really honest with ourselves we all have plenty of time to exercise we just like to think we are 'too' busy because we are unwilling to sacrifice some of our free time (probably spent watching TV or at the pub) so that we can make it a priority.
Exercise doesn't have to be all or nothing and you can still have a life outside of your fitness regime. If you are concerned you will struggle to fit in everything you want to do then try splitting your exercise into chunks. A study from John Hopkins School of Medicine revealed that women who exercised for 10 minutes 3 times a day lost a larger amount of weight than those exercising for 40 minute chunks.
Another tip is to wake up a little bit earlier. Understandably when you come home in the evening you may just want to relax so exercise can seem unappealing and tiresome. Instead why not consider a morning workout, swim, run etc?
According to research those who exercise in the morning are 90 per cent more likely to stick with it and also burn approximately three times more fat. The science behind the research states that because your body burns carbohydrates whilst you sleep, exercise in the morning means it has to tap into fat reserves for energy.
Writing down what you want to achieve increases your chances of reaching your goals. Robert Bray, a personal trainer, has suggested keeping an exercise journal containing realistic goals and their deadlines. It may be that you want to loose a specific amount of weight for your wedding or a holiday, so write this down as an end goal so you are constantly reminded of your motivation.
In addition you could try to recruit a friend. Apparently working out with a friend really enhances your chances of sticking with it and produces a higher number of endorphins than when you exercise on your own.
There is no denying that gyms are expensive and often involve difficult to terminate minimum terms. However, there are plenty of other options which are very low cost or free. For example, skipping is a great calorie burner so purchase an inexpensive skipping rope, pop yourself in front of the TV or in the garden and do five blocks of three minutes skipping (this will burn up to 600 calories an hour). Other good alternatives are power walking, cycling or jogging.
For more tips and the original article please visit All About You.
Why a weekend lie in doesn't help to improve performance Relationship Goals - Step 2 - How to work out what you want from a relationship?


Many committed Europeans have been asking themselves this question in light of increasing Euro-scepticism and national egotisms 60 years after the execution of the Treaties of Rome. In particular growing social inequality is creating divides amongst communities and societies so that ever more people – among them many young people – are acutely endangered by occupational and social exclusion.
Concurrently, for the past 30 years, the Erasmus+ programme has been successfully putting European exchange and learning experiences for young people into action. Programmes like the European Voluntary Service (EVS), new mobility initiatives like the European Solidarity Corps and a range of national mobility programmes are not only popular, but more urgently needed than ever before to make Europe more perceptible, a truly lived experience. And yet, these offers are made use of only by a minority of people! How can EVS and other mobility programmes be made more accessible for young people, regardless of their educational background and social status? What are possible approaches to a European renewal process that promises a socially just and inclusive Europe, above all for young people?
These and other questions will be discussed at the European conference "Volunteering for Social Change" which will be held by the Allianz Cultural Foundation and its partners at the Allianz Forum in Berlin on 20 and 21 March.
The conference will be opened with an evening event on 20 March during which i.a. Daniel Cohn-Bendit and MEP Brando Benifei will discuss which measures must be taken in order to create a socially just and inclusive Europe for young people. On Tuesday, the results of the European cooperation project "European Voluntary Service for All" (EVS4ALL) will be presented. By analysing concrete voluntary services fourteen organisations from the fields of European youth and voluntary work as well as foundations from Germany, France, Great Britain, Poland, Spain and Romania have developed European policy recommendations for an inclusive voluntary service which will be discussed with i.a. representatives of the European Commission. The project is funded by the Erasmus+ programme.
The conference will be translated into German and English. Click here for more information and to view the full conference programme.


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Military Drones Repurposed for Refugee Rescue
Posted By: Miriam McNabbon: October 05, 2015
Austrian military drone manufacturer Schiebel has teamed with refugee rescue organization Migrant Offshore Aid Station (MOAS) to save over 8,800 refugees stranded in the Mediterranean so far this year. MOAS is the first civilian organization to use the Camcopter® S-100 for humanitarian purposes.
MOAS has used the S-100 to assist in search and rescue missions for troubled ships carrying refugees from a variety of countries, including Eritrea, Somalia, and Syria, across the Mediterranean. The S-100's camera delivers daylight and infrared video in real time to the MOAS team: the high quality footage enables the team to locate small ships several miles away and accurately assess the type of boat and it's level of danger. After locating a vessel in distress, MOAS notifies the appropriate official rescue organizations, and then provides additional aid as needed including water, blankets, life jackets, and emergency first-aid treatment.
MOAS' Phoenix, a 40 meter ship on which the military-grade drone is stationed, works in cooperation with Rome's Maritime Rescue Coordination Centre to perform refugee rescues. The drone, largely sponsored by Schiebel, is operated by Schiebel staff. Schiebel plans to continue to support the operation, offering completely free use of the drone in 2015, according to an MOAS spokesperson.
The Migrant Offshore Aid Station, a Malta-based NGO, was formed in 2013 by the Catrambone family following the drowning of 400 migrants near the Italian island of Lampedusa. MOAS patrols major migrant shipping lanes to locate vessels in distress. Previously operating only in the Mediterranean, MOAS plans to expand operations to South East Asia this year.
"The ultimate aim is to mitigate loss of life," says Martin Xuereb, director of MOAS.
Civilian Drones Search and Rescue
A Drone Poem
Drone Attracts Attention
Smart UAVS for Quicker Search and Rescue
Tags: CamcopterMOASRefugeesSchiebelSearch and Rescue Drone


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Home Blog General World Food Café- Volunteers needed
General World Food Cafe
World Food Café- Volunteers needed
Updated on November 17, 2020 August 22, 2017
W.O.M.E.N in Iceland is looking for volunteers for our World Food Cafe events.
What is the World Food Cafe?
The world food cafe is an event held once a month from September through May where volunteers gather with their girlfriends and cook their favourite meals from their homeland to share with the guests of the event. This event is usually held the first Thursday of every month.
The event usually books up quickly and we only allow 30 registered guests. All women are invited to register for the evening, and it is a women only event.
Chef volunteers!
Through the years we have had women living in Iceland share with us their special dishes which they enjoy from their homeland. In the past years we have had tasty food from Turkey, South Korea, Kenya, Mexico, Scotland to name a few. If you have cooked your cuisine at our event and would like to come back and share more dishes, please do!
If you and your girlfriends have interest in cooking for our event and sharing some of your favourite meals from homeland please contact us and we can send you more information.
Assistants needed!
We are also looking for women to volunteer and be willing to help on the day of the event. They would have to be available from 17:00 to 22:00 the evening of the event. Volunteer assistants do not pay for the meal that will be served the night of the event. Volunteer assistants will help set up the hall for the event, assist during the event and help to straighten up the hall after the event. You will NOT be expected to do dishes.
Contact us at eldhus@womeniniceland.is if you are interested in cooking or assisting at our monthly event.
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HOLLYWOOD, Fl – Artist Michelle Weinberg unveiled her proposed designs for sidewalk artworks for six neighborhoods in Hollywood, Florida. The tile mosaics will be approximately 27 feet by 1 foot and applied to new sidewalks near key community facilities. The unveiling was at the Art and Culture Center of Hollywood on August 7, 2013. City Commissioners Richard Blattner, Traci Callari and Peter Hernandez attended along with City Manager Cathy Swanson-Rivenbark.
Michelle Weinberg presented her work at the Art and Culture Center of Hollywood. Art and Culture Center Executive Directer, Joy Satterlee (left) stands besides Ms. Weinberg (right).
Click here for all posts about the Hollywood sidewalks.