text:
\section{Introduction}
Nowadays the study of higher-order theories is an interesting subject in theoretical physics. In fact, since the works developed by Ostrogradski concerning the  Hamiltonian formulation of such systems \cite{1}  up to now where  recent works regarding systems just like generalizations of electrodynamics \cite{2, 3, 4, 5}, generalized meson-field theory \cite{5a, 5b}, string theory \cite{6a, 6}, dark energy physics \cite{7, 8} have  been analyzed. Moreover, from the gravitational point of view, we find interesting models of higher derivative gravity where  are useful for testing the investigation of quantum gravitational effects. Those models are formed with the Einstein-Hilbert action and the addition of quadratic products of the curvature tensor; the main attractive feature of those models is the renormalizability \cite{9, 10}.
Within the context of three dimensions, we find an interesting and natural higher-order model, the so-called topologically massive gravity [TMG] \cite{11a}. This theory is  the coupling of Einstein-Hilbert action plus a Chern-Simons term, at a linear level it describes the propagation of a single massive state of helicity $\pm$ 2 on a Minkowski background, the so-called massive graviton. The theory is a good laboratory for testing classical and quantum ideas of gravity because it is the unique local dynamical and unitary gravity model which is naive power counting renormalizable \cite{11c}. In this respect, the study of  higher-order field theories is carried out by using the so-called Ostrogradski-Dirac framework \cite{11b}. In fact, the Ostrogradski-Dirac framework is based on the extension of the phase space, where  the choice of the fields and their temporal derivatives are the canonical variables. In other words, a generalization of the canonical momenta  is introduced, then the identification of the constraints is performed as in Dirac's method is done \cite{11}. However, in some cases it is claimed that the usual Ostrogradski-Dirac framework does not have the control for identifying the constraints in consistent form, then the constraints are fixed by hand \cite{12a}. In this respect, a detailed Hamiltonian formulation should provide a complete description of the system, and the complete set of either constraints or gauge invariance transformations must be correctly identified. Hence, it is mandatory to perform a  correct analysis  of the constraints in any   canonical formulation. In this respect, with all constraints at hand and classified into first class and second class, the Dirac brackets, useful to quantize a gauge system can be constructed, then the second-class constraints and non-physical degrees of freedom are removed. In the case of higher-order gravity theories, the separation of the constraints into first-and second-class is a delicate issue, therefore, alternative approaches can be required. In this respect, there exists another, alternative, a canonical formulation of higher-order systems the so-called Gitman-Lyakhovich-Tyutin [GLT] formalism \cite{14, 15}. The GLT formalism is a generalization of Ostrogradski's framework that is based on the introduction of extra fields reducing a problem with higher derivatives to one with first-order time derivative, then by using either a correct gauge fixing and the introduction of the Dirac brackets, the second class constraints and non-physical degrees of freedom can be removed. \\
In this manner, with all commented above, the purpose of the present work is to report a detailed GLT study of TMG theory. Our analysis will follow a different procedure to that presented in \cite{15a}. We  will develop a detailed GLT framework as an alternative study beyond Ostrogradski-Dirac framework. In fact, the correct canonical analysis of a given classical theory is the first step towards its canonical quantization and so it is worthwhile to perform it. The  complete structure of the constraints without fixing them by hand  are reported. In addition, we will compare our results with those reported in the literature.\\
The paper is organized as follows. In Section II the GLT method for higher-order Chern-Simons theory is developed. We report the complete set of the constraints and the Dirac brackets are constructed; the Dirac algebra between the constraints and the first-class extended Hamiltonian is reported. In Section III the TMG theory is analyzed. We perform a detailed GLT formalism and the new structure of the constraints is reported. In addition, the Dirac brackets and the algebra of the constraints are calculated.

\section{GLT analysis of higher order Chern-Simons term}
We start with the standard linear  form of the Chern-Simons Lagrangian density  given by \cite{15a}
\begin{equation}
    L_{CS}=\frac{1}{2}\epsilon^{\lambda \mu \nu} (\partial_\sigma h^{\rho}{}_{\lambda} \partial_\rho \partial_\mu h^{\sigma}{}_{\nu}-\partial_\sigma h^{\rho}{}_{\lambda} \partial^\sigma \partial_\mu h_{\rho \nu}),
\label{eq:lcs}
\end{equation}
here, spacetime indices are represented by  the greek alphabet $\alpha, \beta = 0, 1, 2$ and space indices  by the latin   $i, j, k =1, 2$, $h_{\mu \nu}$ is the perturbation of the metric around the flat spacetime geometry and the following signature  $\eta_{\mu \nu}= (-1, 1, 1)$ is used. By performing the $2+1$ decomposition, we can write the action as
\begin{equation}
\begin{aligned}
    L_{CS}= {} & - \epsilon^{ij}(\partial^k h_{0j}\Ddot{h}_{ki} + \partial_j h^{k}{}_{0}\Ddot{h}_{ki} - \frac{1}{2} \dot{h}^{k}{}_{j}\Ddot{h}_{ki} + \partial_j \partial^k h_{00} \dot{h}_{ki}+                \partial^k \partial_i h_{0j} \dot{h}_{k0} \\
               & + \frac{1}{2} \nabla^2 h_{0j} \dot{h}_{0i} - \frac{1}{2} \nabla^2 h^{k}{}_{j} \dot{h}_{ki} + \frac{1}{2} \partial_k \partial_l h^{l}{}_{j} \dot{h}^{k}{}_{i} + \nabla^2 h_{00} \partial_i h_{0j} \\
               & - \nabla^2 h^{k}{}_{0} \partial_i h_{kj} - \partial^l \partial_i h^{k}_{j} \partial_k h_{l0}).
\end{aligned}
\label{eq:cssep}
\end{equation}
We can observe that the Lagrangian is a higher-order derivative theory and the standard way for performing the canonical analysis is by using  the Ostrogradski-Dirac framework, however, we have commented above the interest to develop an alternative GLT analysis. It is worth commenting that our results are new and are not reported in the literature.  For our aims, we need to rewrite the Lagrangian by introducing the following variables
\begin{equation}
    G_{\mu \nu} = \dot h_{\mu \nu}
 \quad \quad
    v_{\mu \nu} = \dot G_{\mu \nu}=\Ddot{h}_{\mu \nu},
\label{eq:var}
\end{equation}
hence, inserting   (\ref{eq:var}) into  (\ref{eq:cssep}) and introducing the momenta $(\pi ^{\mu \nu},  P^{\mu \nu})$  canonically conjugate  to $(h_{\mu \nu}, G_{\mu \nu})$, we redefine the action as \cite{14, 15}

\begin{equation}
    S= \int{\left[ L_{CS}^{v} + \pi ^{\mu \nu}(\dot h_{\mu \nu} - G_{\mu \nu}) + P^{\mu \nu}(\dot G_{\mu \nu} - v_{\mu \nu})\right]{d}^{3}x},
\label{eq:actioncs}
\end{equation}
where
\begin{equation}
\begin{aligned}
    L_{CS}^{v}= {} & - \epsilon^{ij}(\partial^k h_{0j} v_{ki} + \partial_j h^{k}{}_{0} v_{ki} - \frac{1}{2} G^{k}{}_{j} v_{ki} + \partial_j \partial^k h_{00} G_{ki} + \partial^k \partial_i h_{0j} G_{k0} \\
               & + \frac{1}{2} \nabla^2 h_{0j} G_{0i} - \frac{1}{2} \nabla^2 h^{k}{}_{j} G_{ki} + \frac{1}{2} \partial^k \partial_l h^{l}{}_{j} G_{ki} + \nabla^2 h_{00} \partial_i h_{0j} \\
               & - \nabla^2 h^{k}{}_{0} \partial_i h_{kj} - \partial^l \partial_i h^{k}{}_{j} \partial_k h_{l0}),
\end{aligned}
\label{eq:lagcst}
\end{equation}
we can observe that there is an advantage of the action  (\ref{eq:actioncs}) because it contains only first-order time derivatives of the fields. It is worth mentioning that the introduction of the momenta allows us to identify more easily the constraints in comparison with Ostrogradski's formalism \cite{14, 15}. In fact, in GLT framework it is not necessary the introduction of a generalized canonical momenta for the fields with  higher-order time derivative terms just as in Ostrogradski is done. Furthermore, there  is not any  effect in the counting of degrees of freedom because  (\ref{eq:cssep}) and (\ref{eq:actioncs}) are equivalent (see the appendix A). In this respect, the fundamental Poisson brackets are given by
\begin{eqnarray}
\nonumber
    \left\lbrace h_{\alpha \beta},\pi^{\mu \nu}\right\rbrace &=&\frac{1}{2}\left(\delta^{\mu }{}_{\alpha} \delta^{\nu }{}_{\beta} + \delta^{\mu }{}_{\beta} \delta^{\nu }{}_{\alpha}\right), \\
    \left\lbrace G_{\alpha \beta},P^{\mu \nu}\right\rbrace&=&\frac{1}{2}\left(\delta^{\mu }{}_{\alpha} \delta^{\nu }{}_{\beta} + \delta^{\mu }{}_{\beta} \delta^{\nu }{}_{\alpha}\right).
\label{eq:eq20}
\end{eqnarray}
In this manner,  the canonical Hamiltonian is defined as usual
\begin{equation}
\begin{aligned}
    H_{CS}={} & \pi^{\mu \nu } G_{\mu \nu } + P^{\mu \nu } v_{\mu \nu } - L^v_{CS}  \\
= {} &  \pi^{00} G_{00} + P^{00} v_{00} + (2\pi^{0k} + \frac{1}{2}\epsilon^{kj}\nabla^2 h_{0j} + \epsilon^{ij}\partial^k \partial_i h_{0j})G_{0k}\\
                & + [\pi^{ki} + \epsilon^{ij}(\partial_j \partial^k h_{00} - \frac{1}{2} \nabla^2 h^{k}{}_{j} + \frac{1}{2} \partial^k \partial_l h^{l}{}_{j})]G_{ki} + 2P^{0i}v_{0i}\\
                & + [P^{ki} + \epsilon^{ij}(\partial^k h_{0j} + \partial_j h^{k}{}_{0} - \frac{1}{2} G^{k}{}_{j} )]v_{ki} \\
                & + \epsilon^{ij} ( \nabla^2 h_{00} \partial_i h_{0j} - \nabla^2 h^{k}{}_{0} \partial_i h_{kj} - \partial^l \partial_i h^{k}{}_{j} \partial_k h_{l0} ),
\end{aligned}
\label{eq:HCS}
\end{equation}
where (\ref{eq:var}) and (\ref{eq:lagcst}) have been used. Thus,  from the  Lagrangian (\ref{eq:lagcst}) we   identify the following primary constraints \cite{14, 15}
\begin{equation}
    Q^{\mu \nu} : \frac {\partial L_{CS}^{v}}{\partial v_{\mu \nu }} - P^{\mu \nu } \approx 0,
\end{equation}
say
\begin{equation}
\begin{aligned}
    Q^{ki} {} & : \frac {\partial L_{CS}^{v}}{\partial v_{ki}} - P^{ki} \approx 0\\ & = - \left \{P^{ki} +\frac{1}{2} \left [ \epsilon^{ij}(\partial^k h_{0j} + \partial_j h^{k}{}_{0} - \frac{1}{2} G^{k}{}_{j} ) + \epsilon^{kj}(\partial^i h_{0j} + \partial_j h^{i}{}_{0} - \frac{1}{2} G^{i}{}_{j} ) \right ] \right \} \approx 0,
\end{aligned}
\label{eq:cntrns}
\end{equation}
\begin{equation}
    Q^{00}: \frac {\partial L_{CS}^{v}}{\partial v_{00}} - P^{00}=0 \Rightarrow P^{00}\approx 0,
\label{eq:Res1}
\end{equation}
\begin{equation}
    Q^{0i}: \frac {\partial L_{CS}^{v}}{\partial v_{0i}} - P^{0i}=0 \Rightarrow P^{0i} \approx 0.
\label{eq:Res2}
\end{equation}
The Poisson algebra between the primary constraints is given by
\begin{equation}
    \left\lbrace Q^{00},Q^{00} \right\rbrace = \left\lbrace Q^{00},Q^{0i}\right\rbrace = \left\lbrace Q^{0i},Q^{0i} \right\rbrace \approx 0,
\label{PoissonQQ}
\end{equation}
\begin{equation}
     \left\lbrace Q^{ki},Q^{lm} \right\rbrace=\frac{1}{4} \left (  \epsilon^{mi}\eta^{lk} + \epsilon^{li}\eta^{mk} + \epsilon^{lk}\eta^{mi} + \epsilon^{mk}\eta^{li} \right ),
\end{equation}
and the primary Hamiltonian is defined by
\begin{equation}
    H^{1}_{CS}:=H_{CS}+ \Delta_{\mu \nu}Q^{\mu \nu},
\label{eq:H1}
\end{equation}
where  \(\Delta_{\mu \nu}\) are idetified  as Lagrange multipliers. In this manner, from consistency of primary constraints
\begin{equation}
    \dot Q^{\alpha \beta}=\left\lbrace Q^{\alpha \beta},H_{CS}^{1} \right\rbrace=\left\lbrace Q^{\alpha \beta},H_{CS} \right\rbrace + \Delta_{\mu \nu}\left\lbrace Q^{\alpha \beta},Q^{\mu \nu} \right\rbrace \approx 0,
\label{eq:cnsevol}
\end{equation}
we identify the following secondary constraints
\begin{equation}
   S^{0}: \left\lbrace Q^{00},H_{CS} \right\rbrace= \pi^{00}\approx 0 ,
\end{equation}
\begin{equation}
   S^{i}: \left\lbrace Q^{0i},H_{CS} \right\rbrace=  \left ( \pi^{0i} + \frac{1}{4}\epsilon^{ij}\nabla^2 h_{0j} + \frac{1}{2}\epsilon^{lj} \partial^i \partial_l h_{0j} \right ) \approx 0,
\end{equation}
and relations between the  Lagrange multipliers
\begin{equation}
\begin{split}
   \left\lbrace Q^{ki},H_{CS} \right\rbrace  & = \frac{1}{2} \bigg [ \epsilon^{ij} \left (\partial_j \partial^k h_{00} - \frac{1}{2} \nabla^2 h^{k}{}_{j} + \frac{1}{2} \partial^k \partial_r h^{r}{}_{j} \right ) + \epsilon^{kj} \left ( \partial_j \partial^i h_{00} - \frac{1}{2} \nabla^2 h^{i}{}_{j} + \frac{1}{2} \partial^i \partial_r h^{r}{}_{j} \right ) \bigg ] \\ &\quad  + \pi^{ik} - \frac{1}{2} \left ( \epsilon^{lk}v^{i}_{l} + \epsilon^{li}v^{k}_{l} \right) - \frac{1}{2} \bigg [ \epsilon^{ij} \left ( \partial^{k} G_{0j}+\partial_{j} G^{k}_{0} \right ) + \epsilon^{kj} \left ( \partial^{i} G_{0j}+\partial_{j} G^{i}_{0} \right ) \bigg ] \\   &\quad - \frac{1}{2} \left (  \epsilon^{mk} \Delta^{i}_{m} + \epsilon^{mi} \Delta^{k}_{m}  \right ) \approx 0.
\end{split}
\label{eq:evolqik}
\end{equation}
The Poisson algebra between the secondary constraints is given by
\begin{equation}
    \left\lbrace S^{0},S^{0} \right\rbrace =
     \left\lbrace S^{\alpha},Q^{00} \right\rbrace = \left\lbrace S^{\alpha},Q^{0i}\right\rbrace = \left\lbrace S^{0},Q^{ik}\right\rbrace = 0,
\label{PoissonSS}
\end{equation}
thus, consistency of the secondary constraints allows us to  identify the following tertiary  constraint
 \begin{equation}
\dot S^{0} \rightarrow    \tilde{S}^0 : \left [ \epsilon^{ij} \partial_{j} \partial^{k} G_{ki} + \epsilon^{ij} \nabla^{2} ( \partial_{i} h_{0j})  \right ]\approx0,
\end{equation}
and  $\dot S^{i} $  provides  relations between the Lagrange multipliers
\begin{equation}
\begin{split}
\dot S^{i} \rightarrow    \tilde{S}^i :&  \frac{\epsilon^{ik}}{2} \left (\nabla^{2} G_{0k} + \partial^{l} \partial_{k} G_{0l} - \partial^{l}v_{lk} - \partial_{k}\nabla^{2}h_{00} \right ) \\ &\quad + \frac{\epsilon^{kj}}{2} \left ( \partial_{j} v^{i}{}_{k} + \nabla^{2}\partial_{k}h^{i}{}_{j} - \partial_{l} \partial^{i}\partial_{k}h^{l}{}_{j} + \partial^{i}\partial_{k}G_{0j} \right ) +\frac{1}{2} \left( \epsilon^{ki}\partial^{l}\Delta_{lk} + \epsilon^{kj}\partial_{j}\Delta^{i}{}_{k}  \right) \approx0.
\end{split}
\label{eq:evolsi}
\end{equation}
From temporal evolution  of the tertiary constraint $\tilde{S}^0$,    we do not obtain more constraints because the Lagrange multipliers are mixed
\begin{equation}
    \dot {\tilde{S}}^{0} = \epsilon^{ij} \left ( \partial_{j}\partial^{k}v_{ki} + \nabla^{2}\partial_{i}G_{0j} + \partial_{j}\partial^{k}\Delta_{ki} \right)\approx 0.
\end{equation}
On the other hand,  this is not the end,  because there are more constraints. In fact, we can observe that  the trace of  (\ref{eq:evolqik})  eliminate the Lagrange multipliers and therefore, we identify   other   constraint
\begin{equation}
V=\pi^i{_{i}} +\frac{\epsilon^{ij}}{2} \partial_i \partial_lh^l{_{j}} \approx 0.
\label{60c}
\end{equation}
Moreover, from (\ref{eq:evolqik})  and  (\ref{eq:evolsi}) we eliminate the Lagrange multipliers and other  constraints are obtained
\begin{equation}
\begin{split}
    \tilde{V}^{i} & = \partial_{k}\pi^{ik} + \frac{\epsilon^{kl}}{4}\nabla^2\partial_kh^i{_{l}}- \frac{\epsilon^{kj}}{4}\partial_k \partial^i\partial_lh^l{_{j}} \approx 0.
\end{split}
\label{59c}
\end{equation}
Finally, we observe that the trace of  (\ref{eq:cntrns})  implies other constraint and from consistency  do not emerge further constraints, say
\begin{eqnarray}
        & \eta_{ik}Q^{ik}& = P^{i}{}_{i}, \nonumber \\
        &\dot P^{i}{}_{i}& = \left\lbrace P^{i}{}_{i},H_{CS} \right\rbrace = \tilde{V} \approx 0.
\end{eqnarray}
Therefore the complete set of constraints is given by
\begin{eqnarray}
&Q^{00}&:P^{00} \approx0, \nonumber \\
&Q^{0i}&:P^{0i} \approx0, \nonumber \\
&Q^{ik}&:P^{ki} +\frac{\epsilon^{ij}}{2}  \partial^k h_{0j} + \frac{ \epsilon^{ij}}{2}\partial_j h^{k}{}_{0} - \frac{\epsilon^{ij}}{4} G^{k}{}_{j}  + \frac{\epsilon^{kj}}{2} \partial^i h_{0j} + \frac{\epsilon^{kj}}{2}\partial_j h^{i}{}_{0} - \frac{\epsilon^{kj}}{4} G^{i}{}_{j} \approx0,  \nonumber \\
&S^{0}& :   \pi^{00}  \approx 0, \nonumber \\
&S^{i}&:    \pi^{0i} + \frac{\epsilon^{ij}}{4}\nabla^2 h_{0j} + \frac{\epsilon^{lj}}{2}\partial^i \partial_l h_{0j}  \approx 0, \nonumber\\
 &\tilde{S}^0& : \epsilon^{ij} \partial_{j} \partial^{k} G_{ki} + \epsilon^{ij} \nabla^{2}  \partial_{i} h_{0j} \approx 0, \nonumber \\
  &\tilde{V}^{i} & :\partial_{k}\pi^{ik} + \frac{\epsilon^{kl}}{4}\nabla^2\partial_kh^i{_{l}}- \frac{\epsilon^{kj}}{4}\partial_k \partial^i\partial_lh^l{_{j}} \approx 0, \nonumber \\
 & \tilde{V}&:\pi^i{_{i}} +\frac{\epsilon^{ij}}{2} \partial_i \partial_lh^l{_{j}} \approx 0, \nonumber \\
 & U&: P^{i}{}_{i} \approx 0.
 \label{fullcons}
\end{eqnarray}
Now, the  not trivial algebra between all constraints is expressed by
\begin{equation}
W = \bordermatrix{
                   & Q^{00}    & Q^{0l}   & Q^{lm}   & S^{ 0}   & S^{l}   & \tilde{S}^{0}   & \tilde{V}^{l}   & \tilde{V}   &U   \cr
    Q^{00}         & 0         & 0        & 0        & 0        & 0        & 0               & 0               & 0        & 0        \cr
    Q^{0i}         & 0         & 0        & 0        & 0        & 0        & 0               & 0               & 0        & 0        \cr
    Q^{ik}         & 0         & 0        & \left\lbrace Q^{ik},Q^{lm} \right\rbrace        & 0        & \left\lbrace Q^{ik},S^{l} \right\rbrace & \left\lbrace Q^{ik},\tilde{S}^{0} \right\rbrace                & 0               & 0         & 0       \cr
    S^{ 0}         & 0         & 0        & 0        & 0        & 0        & 0               & 0               & 0         & 0       \cr
    S^{ i}         & 0         & 0        & \left\lbrace S^{i},Q^{lm} \right\rbrace        & 0        & \left\lbrace S^{i},S^{l} \right\rbrace        & \left\lbrace S^{i},\tilde{S}^{0} \right\rbrace               & 0               & 0         & 0       \cr
    \tilde{S}^{0}  & 0         & 0        & \left\lbrace \tilde{S}^{0},Q^{lm} \right\rbrace        & 0        & \left\lbrace \tilde{S}^{0},{S}^{l} \right\rbrace         & 0               & 0               & 0         & 0       \cr
    \tilde{V}^{i}  & 0         & 0        & 0        & 0        & 0        & 0               & 0            & 0           & 0     \cr
    \tilde{V}      & 0         & 0        & 0        & 0        & 0        & 0               & 0            & 0           & 0     \cr
    U              & 0         & 0        & 0        & 0        & 0        & 0               & 0            & 0           & 0     \cr
            }.
    \label{eq:WCS}
\end{equation}
where
\begin{eqnarray}
& \left\lbrace Q^{ik},Q^{lm} \right\rbrace & = \frac{1}{4} \left (  \epsilon^{mi}\eta^{lk} + \epsilon^{li}\eta^{mk} + \epsilon^{lk}\eta^{mi} + \epsilon^{mk}\eta^{li} \right )\delta^{2}(x-y), \nonumber \\
& \left\lbrace Q^{ik},S^{l} \right\rbrace & =  \frac{1}{4} \left ( \epsilon^{il}\partial^{k} + \epsilon^{kl}\partial^{i} + \epsilon^{ij}\eta^{kl}\partial_{j} + \epsilon^{kj}\eta^{il}\partial_{j} \right )\delta^{2}(x-y), \nonumber \\
& \left\lbrace Q^{ik},\tilde{S}^{0} \right\rbrace & = \frac{1}{2} \left ( \epsilon^{ji}\partial_{j}\partial^{k} + \epsilon^{jk}\partial_{j}\partial^{i} \right ) \delta^{2}(x-y), \nonumber \\
& \left\lbrace S^{i},S^{l} \right\rbrace & = \frac{1}{4} \left ( \epsilon^{il}\nabla^{2} + \epsilon^{ij}\partial^{l}\partial_{j} + \epsilon^{jl}\partial^{i}\partial_{j} \right )\delta^{2}(x-y), \nonumber \\
& \left\lbrace S^{i},\tilde{S}^{0} \right\rbrace & = \frac{1}{2}\epsilon^{li}\nabla^{2}\partial_{l}\delta^{2}(x-y),
\end{eqnarray}
from the matrix (\ref{eq:WCS})  we can classify  the constraints into first class and second class. In fact,  from  the rank of this matrix  we identify the set of second class constraints,  and  the  null vectors allows us to  identify the first class constraints  \cite{18}, thus,  from the null vectors  we identify the following first class  constraints
    \begin{eqnarray}
&Q^{00}&:P^{00} \approx0, \nonumber \\
&Q^{0i}&:P^{0i} \approx0, \nonumber \\
&S^{0}& :   \pi^{00}  \approx 0, \nonumber \\
&S^{i}&:   \partial_{k}P^{ki} - \pi^{0i} + \frac{\epsilon^{ij}}{2}\partial_{k}\partial_{j}h^{k}{}_{0} +  \frac{\epsilon^{ij}}{4}\nabla^{2}h_{0j} - \frac{\epsilon^{ij}}{4}\partial^{k}G_{kj} - \frac{\epsilon^{kj}}{4}\partial_{k}G^{i}_{j} \approx0 \nonumber\\
&\tilde{V}^{i} & :\partial_{k}\pi^{ik} + \frac{\epsilon^{kl}}{4}\nabla^2\partial_kh^i{_{l}}- \frac{\epsilon^{kj}}{4}\partial_k \partial^i\partial_lh^l{_{j}} \approx 0, \nonumber \\
 & \tilde{V}&:\pi^i{_{i}} +\frac{\epsilon^{ij}}{2} \partial_i \partial_lh^l{_{j}} \approx 0, \nonumber \\
 & U&: P^{i}{}_{i} \approx 0,\nonumber \\
 &\tilde{S}^0& \partial_{i}\partial_{k}P^{ik} + \frac{\epsilon^{ij}}{2} \partial_{i} \partial^{k} G_{kj},
\end{eqnarray}
and the rank implies  the following two  second class constraints
    \begin{eqnarray}
& Q^{11}& = P^{11} + \partial^{1}h_{02} + \partial_{2}h^{1}{}_{0} -\frac{1}{2}G^{1}{}_{2} \approx 0, \nonumber\\
& Q^{12}& = P^{12} + \partial^{2}h_{02} - \partial^{1}h_{01} + \frac{1}{4}G^{1}{}_{1} - \frac{1}{4}G^{2}{}_{2}  \approx 0,
    \end{eqnarray}
we can observe that the constraints are identified by means of the rank and the null vectors;  it is not necessary to fixing  them by hand such as in Ostrogradski formalism is done \cite{15a}.  For instance, one null vector is given by $w=(0, 0,  \partial_i \omega, 0, \eta_{i}{^{j} }\omega, 0, 0, 0)$, where $\omega$ is an arbitrary function. From the contraction of the vector $w$  with the constraints (\ref{fullcons}), the first class constraint   $S^i$  is obtained. \\
With the correct constraints,  we are able to calculate the number of physical degrees of freedom. In fact, there are 24 canonical variables, 11 first class constraints and two second class constraints, this allows us to conclude that
\begin{equation}
DOF = \frac{1}{2}\left[ \left ( 24 \right ) -2\left ( 11\right )- \left ( 2\right )  \right] = 0,
\end{equation}
as expected.\\
To complete the analysis, it is well-known that for two functions on the phase space, namely $A$ and $B$, the Dirac bracket between these variables is defined by
\begin{equation}
    \left\lbrace A,B \right\rbrace_{D} =  \left\lbrace A,B \right\rbrace - \int dudv  \left\lbrace A,\chi^{\alpha} (u) \right\rbrace C^{-1}_{\alpha \beta} \left\lbrace \chi^{\beta} (v),B \right\rbrace,
    \label{eq:DIRACB}
\end{equation}
where we have noted to  $\chi^{\alpha}$ and  $\chi^{\beta}$ as the second class constraints,   and  $C^{-1}_{\alpha \beta}$  is the inverse matrix of $C^{\alpha\beta}$ whose entries are the Poisson brackets between the second class constraints. Thus, for the theory under study  $C^{\alpha\beta}$  takes the form
\begin{equation}
    C^{\alpha \beta} = \begin{pmatrix}
        \left\lbrace Q^{11},Q^{11} \right\rbrace &\left\lbrace Q^{11},Q^{12} \right\rbrace  \\
        \left\lbrace Q^{12},Q^{11} \right\rbrace & \left\lbrace Q^{12},Q^{12} \right\rbrace
    \end{pmatrix} = \begin{pmatrix}
        0 & -\frac{1}{2} \\
        \frac{1}{2} & 0
    \end{pmatrix}\delta^{2}(x-y),
\end{equation}
and its inverse is given by
\begin{equation}
    C^{-1}_{\alpha \beta} = \begin{pmatrix}
        0  & 2 \\
        -2 & 0
    \end{pmatrix}\delta^{2}(x-y).
\end{equation}
Therefore, the non trivial  Dirac's  brackets between the canonical variables are given  by
\begin{eqnarray}
&\left\lbrace h_{00},\pi^{00} \right\rbrace_{D}& = \delta^{2}(x-y), \nonumber \\
&\left\lbrace h_{0i},\pi^{0l} \right\rbrace_{D}& = \frac{1}{2}\delta_{l}{}^{i}\delta^{2}(x-y), \nonumber \\
&\left\lbrace h_{ij},\pi^{lm} \right\rbrace_{D}& = \frac{1}{2} \left( \delta_{i}{}^{l}\delta_{j}{}^{m} + \delta_{i}{}^{m}\delta_{j}{}^{l}\right) \delta^{2}(x-y), \nonumber \\
&\left\lbrace \pi^{0i},\pi^{0l} \right\rbrace_{D}& = -\frac{1}{2} \epsilon^{il}\nabla^{2} \delta^{2}(x-y), \nonumber \\
&\left\lbrace \pi^{0i},G_{lm} \right\rbrace_{D}& = \frac{1}{2} \left( \delta_{l}{}^{1}\delta_{m}{}^{2} + \delta_{l}{}^{2}\delta_{m}{}^{1} \right) \left( \epsilon^{1i}\partial^{1} +  \eta^{1i}\partial^{2} \right) \delta^{2}(x-y) \nonumber \\
&& -\frac{1}{2}\delta_{l}{}^{1}\delta_{m}{}^{1} \left( \epsilon^{1i}\partial^{2} +  \epsilon^{2i}\partial^{1} + \eta^{2i}\partial^{2} - \eta^{1i}\partial^{1} \right) \delta^{2}(x-y), \nonumber \\
&\left\lbrace \pi^{0i},P^{lm} \right\rbrace_{D}& = -\frac{1}{8}\left(  \epsilon^{ln}\eta^{im} + \epsilon^{mn}\eta^{il} + \epsilon^{mi}\eta^{ln} + \epsilon^{li}\eta^{mn}  \right)\partial_{n} \delta^{2}(x-y), \nonumber \\
&\left\lbrace G_{00},P^{00} \right\rbrace_{D}& = \delta^{2}(x-y), \nonumber \\
&\left\lbrace G_{0i},P^{0l} \right\rbrace_{D}& = \frac{1}{2}\delta_{i}{}^{l}\delta^{2}(x-y), \nonumber \\
&\left\lbrace G_{ij},G_{lm} \right\rbrace_{D}& = \delta_{i}{}^{1}\delta_{j}{}^{1}\left( \delta_{l}{}^{1}\delta_{m}{}^{2} + \delta_{l}{}^{2}\delta_{m}{}^{1} \right)\delta^{2}(x-y) - \delta_{l}{}^{1}\delta_{m}{}^{1}\left( \delta_{i}{}^{1}\delta_{j}{}^{2} + \delta_{i}{}^{2}\delta_{j}{}^{1} \right)\delta^{2}(x-y), \nonumber \\
&\left\lbrace G_{ij},P^{lm} \right\rbrace_{D}& = \frac{1}{4}\delta_{i}{}^{1}\delta_{j}{}^{1}\left(  \epsilon^{1m}\eta^{2l} + \epsilon^{1l}\eta^{2m} + \epsilon^{2m}\eta^{1l} + \epsilon^{2l}\eta^{1m}\right)\delta^{2}(x-y) \nonumber \\
&& -\frac{1}{4}\left( \delta_{i}{}^{1}\delta_{j}{}^{2} + \delta_{i}{}^{2}\delta_{j}{}^{1}\right)\left( \epsilon^{1m}\eta^{1l} + \epsilon^{1l}\eta^{1m} \right)\delta^{2}(x-y) \nonumber \\
&& +\frac{1}{2} \left( \delta_{i}{}^{l}\delta_{j}{}^{m} + \delta_{i}{}^{m}\delta_{j}{}^{l}\right) \delta^{2}(x-y), \nonumber \\
&\left\lbrace P^{ij},P^{lm} \right\rbrace_{D}& = \frac{1}{16}\left( \epsilon^{il}\eta^{mj} + \epsilon^{im}\eta^{jl} + \epsilon^{jl}\eta^{mi} + \epsilon^{jm}\eta^{il} \right)\delta^{2}(x-y).
\label{dbra}
\end{eqnarray}
Finally, the correct classification of the constraints allows us to construct the extended Hamiltonian. In fact, it is well known that the extended Hamiltonian must to be a first class function. Moreover,  the extended Hamiltonian and the Dirac brackets are the cornerstones for performing the identification of either observables or the study of quantization. The extended Hamiltonian is given by
\begin{equation}
H_{E}=H_{CS}+u_{\alpha} \chi^{\alpha},
\label{HTCS}
\end{equation}
where $u_{\alpha} $ are the Lagrange multipliers that can be obtained from
\begin{equation}
    u_{\alpha}= C^{-1}_{\beta \alpha}\lbrace \chi^{\beta}, H_{CS}\rbrace,
    \label{multi}
\end{equation}
where $\chi^{\alpha}=(\chi^{1},\chi^{2})=(Q^{11},Q^{12})$ are the second class constraints. There are two second class constraints, then there are  two  Lagrange multipliers to be found, say  $(u_1, u_2)$. Hence, by using  (\ref{multi}), the following  multipliers arise
\begin{eqnarray}
    u_{1}&=& C_{21}^{-1}\lbrace \chi^{2},H_{CS}\rbrace \\ \nonumber &=&2\pi^{12}+\partial_{2}\partial_{2}h_{00}-\partial_{1}\partial_{1}h_{00}+\frac{1}{2}\nabla^{2}h^{1}{}_{1}-\frac{1}{2}\nabla^{2}h^{2}{}_{2} +\frac{1}{2}\partial_{l}\partial_{2}h^{l}{}_{2}\nonumber \\ && -\frac{1}{2}\partial_{l}\partial_{1}h^{l}{}_{1} -2\partial_{2}G_{02}+2\partial_{1}G_{01}+v_{22}-v_{11},\\
    u_{2}&=& = C_{12}^{-1} \lbrace \chi^{1},H_{CS}\rbrace \\ \nonumber &=& -2\pi^{11}-2\left( \partial_{2}\partial_{1}h_{00}-\frac{1}{2}\nabla^{2}h_{12}+\frac{1}{2}\partial_{1}\partial_{l}h^{l}{}_{2}\right) +2\partial_{1}G_{02}+2\partial_{2}G_{01} -2v_{21},
\end{eqnarray}
and the extended Hamiltonian  takes the form
\begin{equation}
    H_{ECS}= H_{CS}+u_{\alpha}\chi^{\alpha}=H_{CS}+u_{1}Q^{11}+u_{2}Q^{12}.
    \label{eq:mult1}
\end{equation}
The extended Hamiltonian given in (\ref{eq:mult1}) is a  first class function. In fact,  its  algebra with all  first class constraints is given by
\begin{eqnarray}
     \left\lbrace Q^{00},H_{ECS} \right\rbrace_{D}  &=&-\pi^{00}=-S^{0}, \\
     \left\lbrace Q^{0i},H_{ECS} \right\rbrace_{D} &=& -\delta^{i}{}_{2}\partial_{2}Q^{11}+\delta^{i}{}_{1}\partial_{1}Q^{11}+\delta^{i}{}_{2}\partial_{1}Q^{12}+\delta^{i}{}_{1}\partial_{2}Q^{12}\\ \nonumber && - \pi^{0i} - \frac{1}{4} \epsilon^{ij} \nabla^{2} h_{0j} - \frac{1}{2}\epsilon^{kj}\partial^{i}\partial_{k}h_{0j}= S^{i}, \\
     \left\lbrace S^{0},H_{ECS} \right\rbrace_{D}  &=& \partial_{1}\partial_{1}Q^{11}-\partial_{2}\partial_{2}Q^{11}+2\partial_{1}\partial_{2}Q^{12}-\epsilon^{ij}\partial_{j}\partial^{k}G_{ki}-\epsilon^{ij}\nabla^{2}\partial_{i}h_{0j}=\tilde{S}^{0},\\
     \left\lbrace U,H_{ECS} \right\rbrace_{D}  &=& -\tilde{V},\\
     \left\lbrace \tilde{V}^{i},H_{ECS} \right\rbrace_{D}  &=& 0,\\
     \left\lbrace \tilde{V},H_{ECS} \right\rbrace_{D}  &=& \partial_{1}\partial_{1}Q^{11}-\partial_{2}\partial_{2}Q^{11}+2\partial_{1}\partial_{2}Q^{12}-\epsilon^{ij}\partial_{j}\partial^{k}G_{ki}-\epsilon^{ij}\nabla^{2}\partial_{i}h_{0j}=\tilde{S}^{0},\\
     \left\lbrace \tilde{S}^{0},H_{ECS} \right\rbrace_{D}  &=& 0, \\
     \left\lbrace S^{i},H_{ECS} \right\rbrace_{D}  &=& \tilde{V}^{i},
\end{eqnarray}
where we observe that the algebra is closed. In this manner, we have performed a complete GLT analysis for the Chern-Simons term in the weak field context, and we have shown that the constraints are obtained with a correct structure  from the null vectors. \\
We finish this section with the calculation of the gauge transformations.  For this aim, we use the first class constraints and we define the generator of gauge transformations as \cite{11}
\begin{eqnarray}
\nonumber
G&=& \int \Big\{ \varepsilon_{00}(y)Q^{00}(y)+ \varepsilon_{0i}(y)Q^{0i}(y)+ \varepsilon_{0}(y)S^{0}(y)+\varepsilon_{i}(y)S^{i}(y)+ \tilde{\varepsilon}_i(y)\tilde{V}^{i}(y)+\tilde{\varepsilon}(y)\tilde{V}(y) \\
&+& \psi (y)U(y)+ \tilde{\psi}_0(y) \tilde{S}^0(y) \Big\} d^2y,
\label{gen}
\end{eqnarray}
where $(\varepsilon_{00},\varepsilon_{0i}, \varepsilon_{0}, \varepsilon_{i}, \tilde{\varepsilon}_i, \tilde{\varepsilon},  \psi, \tilde{\psi}_0 )$ are gauge parameters. Hence, the gauge transformations of the canonical variables $h_{\mu \nu}$ are given by
\begin{eqnarray}
\delta{h_{00}}(x)&=&\{ h_{00}(x), G\}_D=\varepsilon_0(x), \nonumber \\
\delta{h_{0i}}(x)&=& \{ h_{0i}(x), G\}_D=-\varepsilon_i(x), \nonumber \\
\delta{h_{ij}}(x)&=& \{ h_{ij}(x), G\}_D= -\frac{1}{2}(\partial_i \tilde{\varepsilon}_j(x)+ \partial_j \tilde{\varepsilon}_i(x))+ \delta_{ij}\tilde{\varepsilon}(x),
\end{eqnarray}
where (\ref{dbra}) have been used. If we fixing  the gauge parameters as $\varepsilon_0=2\partial_0\zeta_0, -\varepsilon_i=\partial_0\zeta_i+\partial_i\zeta_0,  \delta_{ij}\tilde{\varepsilon}= \frac{1}{2}(\partial_i \zeta_j+ \partial_j \zeta_i), -\tilde{\varepsilon}_i=\zeta_i$, then the gauge transformations are given by
\begin{equation}
\delta h_{\mu \nu}= \partial_\mu \zeta_\nu+ \partial_\nu \zeta_\mu,
\end{equation}
it is easy to see that  the action (\ref{eq:lcs}) and the equations of motion (see appendix B eq. (\ref{cot})) are invariant under these gauge transformations.

\section{The GLT analysis for topologically massive gravity}
We have observed in previous sections,  that the GLT analysis for  the  higher-order  derivative Chern-Simons theory allowed us to know the complete structure of the constraints. Hence, in this section, we shall perform the GLT analysis for TMG theory and we will report the complete structure of the constraints; our study will complete  those results reported in \cite{15a}. \\
Now,  the action under study is given by
\begin{equation}
    S[{g_{\mu \nu}}]=\int \Big \{ R\sqrt {-g} + \left[\frac{1}{\mu}\epsilon^{\lambda \mu \nu}\Gamma_{\lambda \sigma}^{\rho} (\partial_\mu \Gamma_{\rho \nu}^{\sigma} + \frac{2}{3} \Gamma_{\mu \xi}^{\sigma}\Gamma_{\nu \rho}^{\xi}) \right]\Big\} dx^3,
    \label{tmg}
\end{equation}
where $R$ is the Ricci tensor, $g$ is the determinant of the metric tensor and $\mu$  is a coupling constant \cite{ 11a}. If we consider the  week gravitational field  formalism,   the action (\ref{tmg}) is reduced  to   the well known Lagrangian for TMG which it is  given by  \cite{15a}
\begin{equation}
\begin{aligned}
    L_{TMG} = {} &  \frac{1}{4}\partial_{\lambda}h_{\mu \nu}\partial^{\lambda}h^{\mu \nu} - \frac{1}{4}\partial_{\lambda}h^{\mu}{}_{\mu}\partial^{\lambda}h^{\nu}{}_{\nu} + \frac{1}{2}\partial_{\lambda}h^{\lambda}{}_{\mu}\partial^{\mu}h^{\nu}{}_{\nu} - \frac{1}{2}\partial_{\lambda}h^{\lambda}{}_{\mu}\partial_{\nu}h^{\nu \mu}
    \\ & + \frac{1}{2 \mu}\epsilon^{\lambda \mu \nu} (\partial_\sigma h^{\rho}{}_{\lambda} \partial_\rho \partial_\mu h^{\sigma}{}_{\nu}-\partial_\sigma h^{\rho}{}_{\lambda} \partial^\sigma \partial_\mu h_{\rho \nu}).
\end{aligned}
\label{eq:LTMG}
\end{equation}
We have commented in previous sections  that  the action (\ref{eq:LTMG}) describes the propagation of a massive graviton on a Minkowski background \cite{11a}, and the analysis of this action always is   an interesting subject for studying due to the closeness with real gravity. \\
In this manner, by performing the 2+1 decomposition  we obtain
\begin{equation}
\begin{aligned}
     L_{TMG} = {} & \frac{1}{4}\dot h_{ij} \dot h^{ij} - \frac{1}{4} \dot h^{i}{}_{i} \dot h^{k}{}_{k} + \dot h_{ij} \partial^{i}h^{0j} - \dot h^{i}{}_{i} \partial^{k} h^{0}{}_{k} - \frac{1}{2}\partial_{i}h_{0j}\partial^{i}h^{0j} \\ & - \frac{1}{4}\partial_{i}h_{jk}\partial^{i}h^{jk} +  \frac{1}{2}\partial_{i}h^{0}{}_{0}\partial^{i}h^{j}{}_{j} + \frac{1}{4}\partial_{i}h^{j}{}_{j}\partial^{i}h^{k}{}_{k} - \frac{1}{2}\partial_{i}h^{ij}\partial_{j}h^{0}{}_{0} \\ & - \frac{1}{2}\partial_{i}h^{ij}\partial_{j}h^{k}{}_{k} + \frac{1}{2}\partial_{i}h_{j0}\partial^{j}h^{0i} + \frac{1}{2}\partial_{i}h_{jk}\partial^{j}h^{ik} \\ & - \frac{1}{\mu}\epsilon^{ij}(\partial^k h_{0j}\Ddot{h}_{ki} + \partial_j h^{k}{}_{0}\Ddot{h}_{ki} - \frac{1}{2} \dot{h}^{k}{}_{j}\Ddot{h}_{ki} + \partial_j \partial^k h_{00} \dot{h}_{ki}+                \partial^k \partial_i h_{0j} \dot{h}_{k0} \\ & + \frac{1}{2} \nabla^2 h_{0j} \dot{h}_{0i} - \frac{1}{2} \nabla^2 h^{k}{}_{j} \dot{h}_{ki} + \frac{1}{2} \partial_k \partial_l h^{l}{}_{j} \dot{h}^{k}{}_{i} + \nabla^2 h_{00} \partial_i h_{0j} \\ & - \nabla^2 h^{k}{}_{0} \partial_i h_{kj} - \partial^l \partial_i h^{k}{}{j} \partial_k h_{l0}).
\end{aligned}
\label{eq:LLTMG}
\end{equation}
By following the GLT formalism, we introduce the variables  (\ref{eq:var}) into  (\ref{eq:LLTMG}) and we obtain
\begin{equation}
\begin{aligned}
    L_{TMG}^{v}= {} & \frac{1}{4}G_{ij} G^{ij} - \frac{1}{4} G^{i}{}_{i} G^{k}{}_{k} + G_{ij} \partial^{i}h^{0j} - G^{i}{}_{i} \partial^{k} h^{0}{}_{k} - \frac{1}{2}\partial_{i}h_{0j}\partial^{i}h^{0j} \\ & - \frac{1}{4}\partial_{i}h_{jk}\partial^{i}h^{jk} +  \frac{1}{2}\partial_{i}h^{0}{}_{0}\partial^{i}h^{j}{}_{j} + \frac{1}{4}\partial_{i}h^{j}{}_{j}\partial^{i}h^{k}{}_{k} - \frac{1}{2}\partial_{i}h^{ij}\partial_{j}h^{0}{}_{0} \\ & - \frac{1}{2}\partial_{i}h^{ij}\partial_{j}h^{k}{}_{k} + \frac{1}{2}\partial_{i}h_{j0}\partial^{j}h^{0i} + \frac{1}{2}\partial_{i}h_{jk}\partial^{j}h^{ik} \\ & - \frac{1}{\mu}\epsilon^{ij}(\partial^k h_{0j} v_{ki} + \partial_j h^{k}{}_{0} v_{ki} - \frac{1}{2} G^{k}{}_{j} v_{ki} + \partial_j \partial^k h_{00} G_{ki} + \partial^k \partial_i h_{0j} G_{k0} \\ & + \frac{1}{2} \nabla^2 h_{0j} G_{0i} - \frac{1}{2} \nabla^2 h^{k}{}_{j} G_{ki} + \frac{1}{2} \partial^k \partial_l h^{l}{}_{j} G_{ki} + \nabla^2 h_{00} \partial_i h_{0j} \\ & - \nabla^2 h^{k}{}_{0} \partial_i h_{kj} - \partial^l \partial_i h^{k}{}_{j} \partial_k h_{l0}).
\end{aligned}
\label{eq:GvTGM}
\end{equation}
hence, the canonical Hamiltonian for the theory under study is given by
\begin{equation}
    \begin{aligned}
    H_{TMG}&={} \pi^{\mu \nu } \dot g_{\mu \nu } + P^{\mu \nu } \dot G_{\mu \nu } - L^v_{TMG} = \pi^{\mu \nu } G_{\mu \nu } + P^{\mu \nu } v_{\mu \nu } - L^v_{TMG} \\
     = {} & \pi^{00}G_{00} + P^{00}v_{00} +2\pi^{0k}G_{0k} + 2P^{0k}v_{0k} + \pi^{ki}G_{ki} + P^{ki}v_{ki} \\ & -\frac{1}{4}G_{ij} G^{ij} + \frac{1}{4} G^{i}{}_{i} G^{k}{}_{k} - G_{ij} \partial^{i}h^{0j} + G^{i}{}_{i} \partial^{k} h^{0}{}_{k} + \frac{1}{2}\partial_{i}h_{0j}\partial^{i}h^{0j} \\ & + \frac{1}{4}\partial_{i}h_{jk}\partial^{i}h^{jk} -  \frac{1}{2}\partial_{i}h^{0}{}_{0}\partial^{i}h^{j}{}_{j} - \frac{1}{4}\partial_{i}h^{j}{}_{j}\partial^{i}h^{k}{}_{k} + \frac{1}{2}\partial_{i}h^{ij}\partial_{j}h^{0}{}_{0} \\ & + \frac{1}{2}\partial_{i}h^{ij}\partial_{j}h^{k}{}_{k} - \frac{1}{2}\partial_{i}h_{j0}\partial^{j}h^{0i} - \frac{1}{2}\partial_{i}h_{jk}\partial^{j}h^{ik} \\ & + \frac{1}{\mu}\epsilon^{ij}(\partial^k h_{0j} v_{ki} + \partial_j h^{k}{}_{0} v_{ki} - \frac{1}{2} G^{k}{}_{j} v_{ki} + \partial_j \partial^k h_{00} G_{ki} + \partial^k \partial_i h_{0j} G_{k0} \\ & + \frac{1}{2} \nabla^2 h_{0j} G_{0i} - \frac{1}{2} \nabla^2 h^{k}{}_{j} G_{ki} + \frac{1}{2} \partial^k \partial_l h^{l}{}_{j} G_{ki} + \nabla^2 h_{00} \partial_i h_{0j} \\ & - \nabla^2 h^{k}{}_{0} \partial_i h_{kj} - \partial^l \partial_i h^{k}{}_{j} \partial_k h_{l0}).
    \end{aligned}
\label{eq:HTMG}
\end{equation}
and the fundamental Poisson brackets between the canonical variables will be
\begin{eqnarray}
\nonumber
    \left\lbrace h_{\alpha \beta},\pi^{\mu \nu}\right\rbrace &=&\frac{1}{2}\left(\delta^{\mu }{}_{\alpha} \delta^{\nu }{}_{\beta} + \delta^{\mu }{}_{\beta} \delta^{\nu }{}_{\alpha}\right), \\
    \left\lbrace G_{\alpha \beta},P^{\mu \nu}\right\rbrace&=&\frac{1}{2}\left(\delta^{\mu }{}_{\alpha} \delta^{\nu }{}_{\beta} + \delta^{\mu }{}_{\beta} \delta^{\nu }{}_{\alpha}\right),
\label{eqbra2}
\end{eqnarray}
thus, from (\ref{eq:GvTGM}) we can identify the following primary constraints \cite{11b, 11}
\begin{equation}
    Q^{00}: \frac {\partial L_{TMG}^{v}}{\partial v_{00}} - P^{00}=0 \Rightarrow P^{00}\approx 0,
\label{eq:PC1}
\end{equation}
\begin{equation}
    Q^{0i}: \frac {\partial L_{TMG}^{v}}{\partial v_{0i}} - P^{0i}=0 \Rightarrow P^{0i} \approx 0,
\label{eq:PC2}
\end{equation}
\begin{equation}
\begin{aligned}
    Q^{ki} {} & : \frac {\partial L_{TMG}^{v}}{\partial v_{ki}} - P^{ki}\\ & = - \left \{P^{ki} +\frac{1}{2\mu{}{}} \left [ \epsilon^{ij}(\partial^k h_{0j} + \partial_j h^{k}{}_{0} - \frac{1}{2} G^{k}{}_{j} ) + \epsilon^{kj}(\partial^i h_{0j} + \partial_j h^{i}{}_{0} - \frac{1}{2} G^{i}{}_{j} ) \right ] \right \} \approx 0.
\end{aligned}
\label{eq:PC3}
\end{equation}
Furthermore, the primary Hamiltonian is given in  usual  way
\begin{equation}
       H^{1}_{TMG} := H_{TMG} + \Delta_{\mu \nu}Q^{{\mu \nu}},
\label{eq:HTMG1}
\end{equation}
where  $\Delta_{\mu \nu}$ are  identified as Lagrange multipliers. Now, from consistency of the primary constraints say
\begin{equation}
    \dot Q^{\alpha \beta}=\left\lbrace Q^{\alpha \beta},H_{TMG}^{1} \right\rbrace=\left\lbrace Q^{\alpha \beta},H_{TMG} \right\rbrace + \Delta_{\mu \nu}\left\lbrace Q^{\alpha \beta},Q^{\mu \nu} \right\rbrace \approx 0.
\label{eq:PCEV}
\end{equation}
we identify the following secondary constraints
\begin{equation}
S^{0}:   \pi^{00}  \approx 0 ,
\label{eq:SC1}
\end{equation}
\begin{equation}
 S^{i}:  \left ( \pi^{0i} + \frac{1}{4\mu}\epsilon^{ij}\nabla^2 h_{0j} + \frac{1}{2\mu}\epsilon^{lj} \partial^i \partial_l h_{0j} \right ) \approx 0,
\label{eq:SC2}
\end{equation}
and the following relations involving Lagrange multipliers
\begin{equation}
    \begin{split}
      S^{ik }& : \pi^{ik} - \frac{1}{2}G^{ik} + \frac{1}{2}\eta^{ik}G^{j}{}_{j} - \frac{1}{2} \left ( \partial^{i}h^{0k}+ \partial^{k}h^{0i} \right ) + \eta^{ik}\partial^{j}h^{0}{}_{j} \\ & + \frac{1}{2\mu} \bigg [ \epsilon^{ij} \left (\partial_j \partial^k h_{00} - \frac{1}{2} \nabla^2 h^{k}{}_{j} + \frac{1}{2} \partial^k \partial_r h^{r}{}_{j} \right ) + \epsilon^{kj} \left ( \partial_j \partial^i h_{00} - \frac{1}{2} \nabla^2 h^{i}{}_{j} + \frac{1}{2} \partial^i \partial_r h^{r}{}_{j} \right ) \bigg ] \\ &\quad - \frac{1}{2\mu} \left ( \epsilon^{lk}v^{i}_{l} + \epsilon^{li}v^{k}_{l} \right) - \frac{1}{2\mu} \bigg [ \epsilon^{ij} \left ( \partial^{k} G_{0j}+\partial_{j} G^{k}_{0} \right ) + \epsilon^{kj} \left ( \partial^{i} G_{0j}+\partial_{j} G^{i}_{0} \right ) \bigg ] \\ &\quad - \frac{1}{2\mu} \left (  \epsilon^{mk} \Delta^{i}_{m} + \epsilon^{mi} \Delta^{k}_{m}  \right ) \approx0.
\end{split}
\label{eq:LMEXP}
\end{equation}
From consistency of  $S^{0}$  we obtain a tertiary  constraint
\begin{equation}
\dot S^{0} \rightarrow    \tilde{S}^0 : \frac{1}{2}\nabla^{2}h^{i}{}_{i} - \frac{1}{2}\partial_{j}\partial_{i}h^{ij} - \frac{1}{\mu}\left [\epsilon^{ij} \partial_{j} \partial^{k} G_{ki} + \epsilon^{ij} \nabla^{2} ( \partial_{i} h_{0j})  \right ]\approx0,
\end{equation}
and  preservation in time of $ S^{i}$   gives more relations involving Lagrange multipliers
\begin{equation}
\begin{aligned}
\dot S^{i} \rightarrow    \tilde{S}^i :&  \frac{1}{2}\partial_{k}G^{ki} - \frac{1}{2}\partial^{i}G^{j}{}_{j} + \frac{1}{2}\nabla^{2}h^{0i} - \frac{1}{2}\partial_{j}\partial^{i}h^{0j} \\ & + \frac{\epsilon^{ik}}{2\mu} \left (\nabla^{2} G_{0k} + \partial^{l} \partial_{k} G_{0l} - \partial^{l}v_{lk} - \partial_{k}\nabla^{2}h_{00} \right ) \\ & + \frac{\epsilon^{kj}}{2\mu} \left ( \partial_{j} v^{i}{}_{k} + \nabla^{2}\partial_{k}h^{i}{}_{j} - \partial_{l} \partial^{i}\partial_{k}h^{l}{}_{j} + \partial^{i}\partial_{k}G_{0j} \right ) +\frac{1}{2\mu} \left( \epsilon^{ki}\partial^{l}\Delta_{lk} + \epsilon^{kj}\partial_{j}\Delta^{i}{}_{k}  \right) \approx0.
\end{aligned}
\label{eq:LMEXP2}
\end{equation}
Furthermore, from  consistency of $ \tilde{S}^0$ we do not obtain further constraints  but more relations between Lagrange multipliers
\begin{equation}
    \dot {\tilde{S}}^{0} = \frac{1}{2}\nabla^{2}G^{i}{}_{i} - \frac{1}{2}\partial^{j}\partial^{i}G_{ij} -\frac{1}{\mu}\epsilon^{ij} \left ( \partial_{j}\partial^{k}v_{ki} + \nabla^{2}\partial_{i}G_{0j} + \partial_{j}\partial^{k}\Delta_{ki} \right) \approx 0,
\end{equation}
Nonetheless, we can eliminate the Lagrange multipliers from (\ref{eq:LMEXP}) and (\ref{eq:LMEXP2}), thus,  a new constraint is obtained
\begin{equation}
\begin{split}
    \tilde{V}^{i} & = \partial_{k}\pi^{ik} + \frac{\epsilon^{kl}}{4\mu}\nabla^2\partial_kh^i{_{l}}- \frac{\epsilon^{kj}}{4\mu}\partial_k \partial^i\partial_lh^l{_{j}} \approx 0,
\end{split}
\end{equation}
 the trace of  (\ref{eq:LMEXP}) remove the Lagrange multipliers and other new  constraint arise
\begin{equation}
    \tilde{V}=\pi^{i}{}_{i} + \frac{1}{2}G^{j}{}_{j} + \partial^{i}h^{0}{}_{i} + \frac{1}{2\mu}\epsilon^{ij}\partial_{i}\partial_{l}h^{l}{}_{j} \approx 0.
\end{equation}
Moreover, from   the trace of  (\ref{eq:PC3}) emerges other constraint and   preservation in time  ends the search of further constraints
\begin{eqnarray}
        & Q^{ik}\eta_{ik}& = P^{i}{}_{i}\approx 0, \nonumber \\
        &\dot P^{i}{}_{i}& = \left\lbrace P^{i}{}_{i},H^{1}_{CS} \right\rbrace = \tilde{V} \approx 0.
\end{eqnarray}
Therefore the complete set of constraints is given by
\begin{eqnarray}
&Q^{00}&:P^{00} \approx0, \nonumber \\
&Q^{0i}&:P^{0i} \approx0, \nonumber \\
&Q^{ik}&:P^{ki} +\frac{\epsilon^{ij}}{2\mu}  \partial^k h_{0j} + \frac{ \epsilon^{ij}}{2\mu}\partial_j h^{k}{}_{0} - \frac{\epsilon^{ij}}{4\mu} G^{k}{}_{j}  + \frac{\epsilon^{kj}}{2\mu} \partial^i h_{0j} + \frac{\epsilon^{kj}}{2\mu}\partial_j h^{i}{}_{0} - \frac{\epsilon^{kj}}{4\mu} G^{i}{}_{j} \approx0,  \nonumber \\
&S^{0}& :   \pi^{00}  \approx 0, \nonumber \\
&S^{i}&:    \pi^{0i} + \frac{\epsilon^{ij}}{4\mu}\nabla^2 h_{0j} + \frac{\epsilon^{lj}}{2\mu}\partial^i \partial_l h_{0j}  \approx 0, \nonumber\\
 &\tilde{S}^0& : \frac{1}{2}\nabla^{2}h^{j}{}_{j} - \frac{1}{2}\partial_{i}\partial_{j}h^{ij} - \frac{1}{\mu}\epsilon^{ij} \partial_{j} \partial^{k} G_{ki} -\frac{1}{\mu} \epsilon^{ij} \nabla^{2}  \partial_{i} h_{0j} \approx 0, \nonumber \\
  &\tilde{V}^{i} & :\partial_{k}\pi^{ik} + \frac{\epsilon^{kl}}{4\mu}\nabla^2\partial_kh^i{_{l}}- \frac{\epsilon^{kj}}{4\mu}\partial_k \partial^i\partial_lh^l{_{j}} \approx 0, \nonumber \\
 & \tilde{V}&:\pi^i{_{i}} +\frac{1}{2}G^{j}{}_{j} + \partial_{i}h^{0i} + \frac{\epsilon^{ij}}{2\mu} \partial_i \partial_lh^l{_{j}} \approx 0, \nonumber \\
 &U&:P^{i}{}_{i} \approx 0.
\end{eqnarray}

Now, we will separate  the constraints  into first class and second class. For this aim, we calculate the matrix, say   $W^{\alpha \beta}$,  whose entries are given by the Poisson brackets between all constraints, this is
\begin{equation}
W = \bordermatrix{
                   & Q^{00}    & Q^{0l}   & Q^{lm}   & S^{ 0}   & S^{ l}   & \tilde{S}^{0}   & \tilde{V}^{l}   & \tilde{V}  &U    \cr
    Q^{00}         & 0         & 0        & 0        & 0        & 0        & 0               & 0               & 0          &0    \cr
    Q^{0i}         & 0         & 0        & 0        & 0        & 0        & 0               & 0               & 0          &0    \cr
    Q^{ik}         & 0         & 0        & \left\lbrace Q^{ik},Q^{lm} \right\rbrace        & 0        & \left\lbrace Q^{ik},S^{l} \right\rbrace & \left\lbrace Q^{ik},\tilde{S}^{0} \right\rbrace                & 0               & \left\lbrace Q^{ik},\tilde{V} \right\rbrace       &U     \cr
    S^{ 0}         & 0         & 0        & 0        & 0        & 0        & 0               & 0               & 0          &0    \cr
    S^{ i}         & 0         & 0        & \left\lbrace S^{i},Q^{lm} \right\rbrace        & 0        & \left\lbrace S^{i},S^{l} \right\rbrace        & \left\lbrace S^{i},\tilde{S}^{0} \right\rbrace               & 0               & \left\lbrace S^{i},\tilde{V} \right\rbrace     &0       \cr
    \tilde{S}^{0}  & 0         & 0        & \left\lbrace \tilde{S}^{0},Q^{lm} \right\rbrace        & 0        & \left\lbrace \tilde{S}^{0},{S}^{l} \right\rbrace         & 0               & 0               & \left\lbrace \tilde{S}^{0},\tilde{V} \right\rbrace      &0     \cr
    \tilde{V}^{i}  & 0         & 0        & 0        & 0        & 0        & 0               & 0            & 0         &0        \cr
    \tilde{V}      & 0         & 0        & \left\lbrace \tilde{V},Q^{lm} \right\rbrace         & 0        & \left\lbrace \tilde{V},S^{l} \right\rbrace         & \left\lbrace \tilde{V},\tilde{S}^{0} \right\rbrace                & 0            & 0        & \left\lbrace \tilde{V},U \right\rbrace        \cr
    U         & 0         & 0        & 0        & 0        & 0        & 0               & 0               &  \left\lbrace U, \tilde{V} \right\rbrace       &0        \cr
            },
            \label{eq:WTMG}
\end{equation}
where the nontrivial brackets are given by
\begin{eqnarray}
& \left\lbrace Q^{ik},Q^{lm} \right\rbrace & = \frac{1}{4\mu} \left (  \epsilon^{mi}\eta^{lk} + \epsilon^{li}\eta^{mk} + \epsilon^{lk}\eta^{mi} + \epsilon^{mk}\eta^{li} \right )\delta^{2}(x-y), \nonumber \\
& \left\lbrace Q^{ik},S^{l} \right\rbrace & =  \frac{1}{4\mu} \left ( \epsilon^{il}\partial^{k} + \epsilon^{kl}\partial^{i} + \epsilon^{ij}\eta^{kl}\partial_{j} + \epsilon^{kj}\eta^{il}\partial_{j} \right )\delta^{2}(x-y), \nonumber \\
& \left\lbrace Q^{ik},\tilde{S}^{0} \right\rbrace & = \frac{1}{2\mu} \left ( \epsilon^{ij}\partial_{j}\partial^{k} + \epsilon^{kj}\partial_{j}\partial^{i} \right ) \delta^{2}(x-y), \nonumber \\
& \left\lbrace Q^{ik},\tilde{V} \right\rbrace & = -\frac{1}{2}\eta^{ik}\delta^{2}(x-y) , \nonumber \\
& \left\lbrace S^{i},S^{l} \right\rbrace & = \frac{1}{4\mu} \left ( \epsilon^{il}\nabla^{2} + \epsilon^{ij}\partial^{l}\partial_{j} + \epsilon^{jl}\partial^{i}\partial_{j} \right )\delta^{2}(x-y), \nonumber \\
& \left\lbrace S^{i},\tilde{V} \right\rbrace & = -\frac{1}{2}\partial^{i}\delta^{2}(x-y) , \nonumber \\
& \left\lbrace \tilde{S}^{0},\tilde{V} \right\rbrace & = \frac{1}{2}\nabla^{2}\delta^{2}(x-y) , \nonumber \\
& \left\lbrace S^{i},\tilde{S}^{0} \right\rbrace & = \frac{1}{2\mu}\epsilon^{ji}\nabla^{2}\partial_{j}\delta^{2}(x-y), \nonumber \\
& \left\lbrace \tilde{V},U \right\rbrace & = \delta^{2}(x-y),
\end{eqnarray}
hence, after a long algebraic manupulations,  we observe that tha matrix (\ref{eq:WTMG})  has a rank= 4 and nine null vectors. From the null vectors we identify the following first class constraints
    \begin{eqnarray}
&Q^{00}&:P^{00} \approx0, \nonumber \\
&Q^{0i}&:P^{0i} \approx0, \nonumber \\
&S^{0}& :   \pi^{00}  \approx 0, \nonumber \\
&S^{i}&:  \partial_{k}P^{ki} - \pi^{0i} + \frac{\epsilon^{ij}}{4\mu}\nabla^{2}h_{0j} + \frac{\epsilon^{ij}}{2\mu}\partial_{j}\partial^{k}h_{k0} - \frac{\epsilon^{ij}}{4\mu}\partial^{k}G_{kj} - \frac{\epsilon^{kj}}{4\mu}\partial_{k}G^{i}_{j} \approx0 \nonumber\\
&\tilde{S}^0& \partial_{i}\partial_{k}P^{ik} + \frac{1}{2}\nabla^{2}h^{j}_{j}-\frac{1}{2}\partial_{i}\partial_{j}h^{ij} + \frac{\epsilon^{ij}}{2\mu}\partial_{i}\partial_{k}G^{k}{}_{j} \approx0 \nonumber\\
&\tilde{V}^{i} & :\partial_{k}\pi^{ik} + \frac{\epsilon^{kl}}{4\mu}\nabla^2\partial_kh^i{_{l}}- \frac{\epsilon^{kj}}{4\mu}\partial_k \partial^i\partial_lh^l{_{j}} \approx 0,
\end{eqnarray}
and the rank  implies  the following four second class constraints
\begin{eqnarray}
& Q^{11}& = P^{11} + \frac{1}{\mu}\partial^{1}h_{02} + \frac{1}{\mu}\partial_{2}h^{1}{}_{0} -\frac{1}{2\mu}G^{1}{}_{2} \approx 0, \nonumber\\
& Q^{12}& = P^{12} + \frac{1}{\mu}\partial^{2}h_{02} - \frac{1}{\mu}\partial^{1}h_{01} + \frac{1}{4\mu}G^{1}{}_{1} - \frac{1}{4\mu}G^{2}{}_{2}  \approx 0, \nonumber \\
 &\tilde{V}&:\pi^i{_{i}} +\frac{1}{2}G^{j}{}_{j} + \partial_{i}h^{0i} + \frac{\epsilon^{ij}}{2\mu} \partial_i \partial_lh^l{_{j}} \approx 0, \nonumber\\
 &U&: P^{i}{}_{i} \approx 0.
\end{eqnarray}
In this manner, the complete set of constraints allow us to carry out the counting of physical degrees of freedom as follows; there are 24 canonical variables $(h_{\mu\nu}, G_{\mu\nu}, \pi^{\mu\nu}, P^{\mu\nu})$,  four  second class constraints $(Q^{11}, Q^{12}, \tilde V, U)$ and nine  first class constraints $(Q^{00}, Q^{0i}, S^{0}, S^{i}, \tilde S^{0}, \tilde V^{i})$, thus
\begin{equation}
DOF= \frac{1}{2}\left[ \left ( 24 \right ) -2\left ( 9\right ) - \left (4\right ) \right] = 1,
\end{equation}
as  expected \cite{10, 11a, 15a}. From the second class constraints we can construct the Dirac brackets between the canonical variables. The nontrivial Dirac's brackets are given by
\begin{eqnarray}
&\left\lbrace h_{00},\pi^{00} \right\rbrace_{D}& = \delta^{2}(x-y), \nonumber \\
&\left\lbrace h_{0i},\pi^{0l} \right\rbrace_{D}& = \frac{1}{2}\delta_{l}{}^{i}\delta^{2}(x-y), \nonumber \\
&\left\lbrace h_{ij},\pi^{lm} \right\rbrace_{D}& = \frac{1}{2} \left( \delta_{i}{}^{l}\delta_{j}{}^{m} + \delta_{i}{}^{m}\delta_{j}{}^{l}\right) \delta^{2}(x-y), \nonumber \\
&\left\lbrace h_{ij},G_{lm} \right\rbrace_{D}& = -\eta_{ij}\eta_{lm}\delta^{2}(x-y), \nonumber \\
&\left\lbrace \pi^{0i},\pi^{0l} \right\rbrace_{D}& = -\frac{1}{2\mu} \epsilon^{il}\nabla^{2} \delta^{2}(x-y), \nonumber \\
&\left\lbrace \pi^{0i},G_{lm} \right\rbrace_{D}& = \frac{1}{2}\left( \delta^{i}{}_{m}\partial_{l}+\delta^{i}{}_{l}\partial_{m} \right)\delta^{2}(x-y), \nonumber \\
&\left\lbrace \pi^{0i},P^{lm} \right\rbrace_{D}& = -\frac{1}{8\mu}\left(  \epsilon^{ln}\eta^{im} + \epsilon^{mn}\eta^{il} + \epsilon^{mi}\eta^{ln} + \epsilon^{li}\eta^{mn}  \right)\partial_{n} \delta^{2}(x-y), \nonumber \\
&\left\lbrace \pi^{ij},G_{lm} \right\rbrace_{D}& = \frac{1}{4\mu}\eta_{lm}\left( \epsilon^{nj}\partial_{n}\partial^{i}+\epsilon^{ni}\partial_{n}\partial^{j} \right)\delta^{2}(x-y), \nonumber \\
&\left\lbrace G_{00},P^{00} \right\rbrace_{D}& = \delta^{2}(x-y), \nonumber \\
&\left\lbrace G_{0i},P^{0l} \right\rbrace_{D}& = \frac{1}{2}\delta_{i}{}^{l}\delta^{2}(x-y), \nonumber \\
&\left\lbrace G_{ij},G_{lm} \right\rbrace_{D}& =\frac{\mu}{4}\left(  \epsilon_{il}\eta_{jm}+\epsilon_{im}\eta_{jl}+\epsilon_{jl}\eta_{im}+\epsilon_{jm}\eta_{il}\right)\delta^{2}(x-y) ,\nonumber \\
&\left\lbrace G_{ij},P^{lm} \right\rbrace_{D}& = \frac{1}{4}\left( \delta^{l}{}_{i}\delta^{m}{}_{j}+\delta^{l}{}_{j}\delta^{m}{}_{i}-\eta^{lm}\eta_{ij} \right)\delta^{2}(x-y) \nonumber \\
&\left\lbrace P^{ij},P^{lm} \right\rbrace_{D}& = \frac{1}{16\mu}\left( \epsilon^{il}\eta^{mj} + \epsilon^{im}\eta^{jl} + \epsilon^{jl}\eta^{mi} + \epsilon^{jm}\eta^{il} \right)\delta^{2}(x-y),
\end{eqnarray}
where we can observe that these brackets coincide with those  reported in \cite{15a} where the Ostogradski approach was used. Furthermore, the extended action   is given by
\begin{equation}
    S_{EXT}\left[ h_{\mu \nu}, \pi^{\mu \nu}, G_{\mu \nu}, P^{\mu \nu}, \lambda_{\alpha}, w_{\alpha} \right] = \int \left( \pi^{\mu \nu}\dot{h}_{\mu \nu} + P^{\mu \nu}\Ddot{h}_{\mu \nu}-H_{ETMG}-\lambda_{\alpha}\gamma^{\alpha}-w_{\alpha}\chi^{\alpha}  \right) \,d^{3}x, \nonumber
\end{equation}
where $H_{ETMG}$ is the extended  Hamiltonian, $\gamma^{\alpha}$ are the first class constraints, $\chi^{\alpha}$ are the second  class constraints; $\lambda_{\alpha}$ and $w_{\alpha}$ are Lagrange  multipliers enforcing the first class and second class  constraints respectively. The extended Hamiltonian is given by
\begin{equation}
    H_{ETMG}= H_{TMG}+u_{\alpha}\chi^{\alpha}=H_{TMG}+u_{1}Q^{11}+u_{2}Q^{12}+u_{3}\tilde{V}+u_{4}U,
    \label{eq:mult}
\end{equation}
where the u's are four Lagrange multipliers that must be identified  because they are associated with the four second class constraints. Hence, the $ u$'s are given by
\begin{equation}
    u_{\alpha}= C^{-1}_{\beta \alpha}\lbrace \chi^{\beta}, H_{TMG}\rbrace ,
\label{multi2}
\end{equation}
where $\chi^{\alpha}=(Q^{11},Q^{12},\tilde{V},U)$. In this manner, from (\ref{multi2}) we obtain the following expresions for the   multipliers
\begin{eqnarray}
u_{1}&=& C_{21}^{-1}\lbrace \chi^{1},H_{TMG}\rbrace  \nonumber \\ &=& ( 2\mu\pi^{12} -\mu G^{12} -\mu\partial^{1}h^{02}-\mu\partial^{2}h^{01}+\partial^{2}\partial_{2}h_{00}-\partial^{1}\partial_{1}h_{00}+\frac{1}{2}\nabla^{2}h^{1}{}_{1}-\frac{1}{2}\nabla^{2}h^{2}{}_{2}\ \nonumber \\ && +\frac{1}{2}\partial_{l}\partial^{2}h^{l}{}_{2} -\frac{1}{2}\partial_{l}\partial^{1}h^{l}{}_{1} -2\partial^{2}G_{02}+2\partial^{1}G_{01}+v^{2}{}_{2}-v^{1}{}_{1})\delta^{2}(x-y), \\
u_{2}&=&C_{12}^{-1} \lbrace \chi^{1},H_{TMG}\rbrace + C_{42}^{-1}\lbrace \chi^{4},H_{TMG}\rbrace \nonumber \\ &=& (-2\mu\pi^{11} -\mu G^{22}-2\mu \partial^{2}h^{02}-2\partial_{2}\partial^{1}h_{00}+\nabla^{2}h^{1}{}_{2}-\partial^{1}\partial_{l}h^{l}{}_{2} \nonumber\\&& + 2\partial^{1}G_{02}+2\partial_{2}G^{1}{}_{0} -2v^{1}_{2} +\mu \tilde{V})\delta^{2}(x-y), \\
u_{3}&=& C_{43}^{-1} \lbrace \chi^{4},H_{TMG}\rbrace = -\tilde{V}\delta^{2}(x-y), \\
u_{4}&=&  C_{24}^{-1}\lbrace \chi^{2},H_{TMG}\rbrace +C_{34}^{-1}\lbrace \chi^{3},H_{TMG}\rbrace \nonumber\\ &=&(-\mu\pi^{12} +\frac{\mu}{2} G^{12} +\frac{\mu}{2}\partial^{1}h^{02}+\frac{\mu}{2}\partial^{2}h^{01}-v^{2}_{2}-\partial_{2}\partial^{2}h_{00}-\frac{1}{4}\nabla^{2}h^{1}{}_{1}+\frac{1}{4}\nabla^{2}h^{2}{}_{2} \nonumber \\ && -\frac{1}{4}\partial_{l}\partial^{2}h^{l}{}_{2} +\frac{1}{4}\partial_{l}\partial^{1}h^{l}{}_{1} +2\partial^{1}G_{01}-\frac{1}{\mu}\epsilon^{ij}\partial_{i}\partial^{k}G_{kj}+\frac{1}{\mu}\epsilon^{ij}\partial_{i}\nabla^{2}h_{j0} )\delta^{2}(x-y),
\end{eqnarray}
the substitution of these multipliers in the extended Hamiltonian defines a first class function. In fact, the Dirac  algebra between the first class constraints and the extended Hamiltonian is given by
\begin{eqnarray}
     \left\lbrace Q^{00},H_{ETMG} \right\rbrace_{D} &=&-S^{0}, \\
     \left\lbrace Q^{0i},H_{ETMG} \right\rbrace_{D} &=& S^{i}, \\
     \left\lbrace S^{0},H_{ETMG} \right\rbrace_{D}  &=& \tilde{S}^{0},\\
     \left\lbrace \tilde{V}^{i},H_{ETMG} \right\rbrace_{D}  &=&0 ,\\
     \left\lbrace \tilde{S}^{0},H_{ETMG} \right\rbrace_{D} &=& 0, \\
     \left\lbrace S^{i},H_{ETMG} \right\rbrace_{D}  &=& \tilde V^{i} ,
\end{eqnarray}
therefore, with this result, we have constructed  a first class extended Hamiltonian as expected. We have observed that the constraints  in this work are not the same at all to  those reported in \cite{15a}, however, the results of the Dirac brackets are equivalent. In this manner, we have presented a new alternative canonical analysis for TMG that extends those reported in the literature.

\section{Conclusions}
In this paper, a detailed GLT analysis of higher-order Chern-Simons and TMG theories in the perturbative context has been performed. With respect to higher-order Chern-Simons theory, we have reported all the complete structure of the constraints, as far as we know, a complete analysis of the constraints of this theory in the weak field context has not been reported. We found the extended Hamiltonian and we showed by means the Dirac algebra that it is of first-class. On the other hand, from the analysis of TMG we have reported a new structure of the constraints. We obtained the constraints by means of the null vectors and we showed that our analysis is consistent; we constructed a first-class extended Hamiltonian and the Dirac algebra between it and the first-class constraints is closed. In this manner, we observed that GLT formalism is an elegant and pragmatic scheme for analyzing higher-order theories.\\
It is worth emphasizing, that the GLT framework can be applied to other physical systems. In fact, it can be used  in others theories  where the Ostrogradski framework is not easy to develop. In this sense, the results of this paper are the basis for future works where the GLT framework will show advantages for the analysis of the constraints,  for instance,  in higher order modifications of gravity.  However, all  these ideas are in progress and will be reported soon  \cite{19}.
\section{Appendix A}
In general, the higher order Lagrangian  (\ref{eq:lcs}) has the form
\begin{equation}
L_{CS}=L_{CS}\Big(h_{\alpha\beta}, \partial_0h_{\alpha\beta}, \partial_ih_{\alpha\beta}, \partial_{0}\partial_0h_{\alpha\beta}, 2\partial_{0}\partial_ih_{\alpha\beta}, \partial_{i}\partial_jh_{\alpha\beta}  \Big),
\label{A1}
\end{equation}
hence, the equations of motion obtained from (\ref{A1}) are given by
\begin{eqnarray}
&&\frac{\partial L_{CS}}{\partial h_{\alpha \beta}} -\partial_0 \frac{\partial L_{CS}}{\partial (\partial_0 h_{\alpha \beta})}- \partial_i \frac{\partial L_{CS}}{\partial (\partial_i h_{\alpha \beta})} + \partial_0\partial_0 \frac{\partial L_{CS}}{\partial (\partial_0\partial_0 h_{\alpha \beta})} \nonumber \\
&+&  \partial_0\partial_i \frac{\partial L_{CS}}{\partial (\partial_i\partial_0 h_{\alpha \beta})} + \partial_i\partial_j \frac{\partial L_{CS}}{\partial (\partial_i\partial_j h_{\alpha \beta})}=0.
\label{B1}
\end{eqnarray}
On the other hand, the introduction of the momenta  $(\pi ^{\mu \nu},  P^{\mu \nu})$   allows us to rewrite the Lagrangian  (\ref{A1}) as  (see Eq. (\ref{eq:actioncs}))
  \begin{equation}
  S= \int{\left[ L_{CS}^{v} + \pi ^{\mu \nu}(\dot h_{\mu \nu} - G_{\mu \nu}) + P^{\mu \nu}(\dot G_{\mu \nu} - v_{\mu \nu})\right]{d}^{3}x},
  \label{B2}
  \end{equation}
  where $L_{CS}^{v}= L_{CS}^{v}\Big(h_{\alpha\beta}, G_{\alpha\beta}, \partial_ih_{\alpha\beta}, v_{\alpha\beta}, \partial_{i}G_{\alpha\beta}, \partial_{i}\partial_jh_{\alpha\beta}  \Big)$, $G_{\alpha \beta}= \dot{h}_{\alpha \beta}$ and $\dot{G}_{\alpha \beta}=v_{\alpha \beta}$. Hence, the variation of (\ref{B2}) respect all dynamical variables is given by
  \begin{eqnarray}
  \frac{\delta S}{\delta h_{\alpha \beta}} &=&  \frac{\partial L_{CS}^{v}}{\partial  h_{\alpha \beta}} - \partial_i \frac{\partial L_{CS}^{v}}{\partial (\partial_i h_{\alpha \beta})}+ \partial_i \partial_j \frac{\partial L_{CS}^{v}}{\partial (\partial_i \partial_j h_{\alpha \beta})}-\dot{\pi}^{\alpha \beta}=0,  \label{95}\\
  \frac{\delta S}{\delta G_{\alpha \beta}} &=&   \frac{\partial L_{CS}^{v}}{\partial  G_{\alpha \beta}}- \partial_i \frac{\partial L_{CS}^{v}}{\partial (\partial_i G_{\alpha \beta})}- \pi^{\alpha \beta} - \dot{P}^{\alpha \beta}=0, \label{96}\\
  \frac{\delta S}{\delta v_{\alpha \beta}} &=&  \frac{\partial L_{CS}^{v}}{\partial  v_{\alpha \beta}}- P^{\alpha \beta}=0,  \label{97} \\
  \frac{\delta S}{\delta \pi^{\alpha \beta}} &=& \dot{h}_{\alpha \beta} - G_{\alpha \beta}=0, \label{98}\\
    \frac{\delta S}{\delta P^{\alpha \beta}} &=& \dot{G}_{\alpha \beta}- v_{\alpha \beta}=0. \label{99}
  \end{eqnarray}
  The equations (\ref{95}-\ref{99}) are equivalent to (\ref{B1}). In fact, from time differentiation   of (\ref{97}) and taking  (\ref{96}) into account,   we obtain
  \begin{equation}
  \frac{\partial L_{CS}^{v}}{\partial  G_{\alpha \beta}}- \partial_i \frac{\partial L_{CS}^{v}}{\partial (\partial_i G_{\alpha \beta})}- \pi^{\alpha \beta}- \partial_0 \frac{\partial L_{CS}^{v}}{\partial v_{\alpha \beta}}=0,
  \label{100}
  \end{equation}
  now, substituting the  time derivative of (\ref{100})  into (\ref{95}) the following arise
  \begin{eqnarray}
 && \frac{\partial L_{CS}^{v}}{\partial  h_{\alpha \beta}} - \partial_i \frac{\partial L_{CS}^{v}}{\partial (\partial_i h_{\alpha \beta})}+ \partial_i \partial_j \frac{\partial L_{CS}^{v}}{\partial (\partial_i \partial_j h_{\alpha \beta})}-  \partial_0 \frac{\partial L_{CS}^{v}}{\partial  G_{\alpha \beta}}+ \partial_0 \partial_i \frac{\partial L_{CS}^{v}}{\partial (\partial_i G_{\alpha \beta})}+ \partial_0\partial_0 \frac{\partial L_{CS}^{v}}{\partial v_{\alpha \beta}}=0,
 \label{101}
  \end{eqnarray}
  finally, if the solutions of (\ref{98}-\ref{99})  are substituted into (\ref{101}), then (\ref{B1}) and (\ref{101}) are equivalents \cite{22, 23}.
 \section{Appendix B}
In this appendix we will resume the difference  between the Chern-Simons and TMG theories. In fact, the equations of motion obtained from  the Chern-Simons action (\ref{eq:lcs}) are given by
\begin{equation}
C^{(L)}_{\mu \nu }=\epsilon_{\mu}{^{\alpha \beta}\partial_\alpha R^{(L)}_{\beta \nu} } + \epsilon_{\nu}{^{\alpha \beta}} \partial_\alpha R^{(L)}_{\beta \mu}=0,
\label{cot}
\end{equation}
where $C^{(L)}_{\mu \nu }$ is the linear Cotton tensor and $R^{(L)}_{\beta \nu}$ is the linear Ricci tensor given by
\begin{equation}
R^{(L)}_{\mu \nu}=\frac{1}{2} \big( -\Box h_{\mu \nu} - \partial_\mu \partial_\nu h+ \partial^\sigma \partial_\nu h_{\sigma \mu}  + \partial^\sigma \partial_\mu h_{\sigma \nu} \big).
\end{equation}
It is well-known that Cotton tensor vanishes  if and only if
the 3-dimensional metric tensor is conformally flat, and this shows  a  difference between Chern-Simons theory and TMG. In fact,  Chern-Symons theory does not describe propagation of degrees of freedom, there is not gravity. \\
On the other hand,  the equations of motion  obtained from  (\ref{eq:LTMG}) are given by
\begin{equation}
G_{\alpha \nu}^{(L)} + \frac{1}{\mu} C^{(L)}_{\alpha \nu }=0,
\label{lin}
\end{equation}
where $G_{\alpha \nu}^{(L)} = R^{(L)}_{\alpha \nu} - \frac{1}{2}g_{\alpha \nu} R^{(L)}$  is the linear Einstein tensor.  By using the Coulomb gauge $\partial^\alpha h_{\alpha \beta}=0$ and the traceless condition $h{^{\alpha}}_\alpha=0$,  the equation (\ref{lin}) can be reduced to  the branch of the massive sector given by
\begin{equation}
h_{\alpha \nu}+ \frac{1}{\mu} \epsilon_{\alpha}{^{\rho \beta}}h_{\beta \nu}=0,
\label{lin2}
\end{equation}
if we contract the equation  (\ref{lin2}) with $(\eta{^\nu}_{\rho} - \frac{1}{\mu} \epsilon_{\rho}{^{ \alpha \nu}}\partial_\alpha)$ we obtain
\begin{equation}
\big(\Box - \mu^2 \big) h_{\mu \nu}=0.
\label{mass2}
\end{equation}
Equation (96) describes the propagation of one degree of freedom a massive spin-2 graviton with mass  $m=\sqrt{\mu^2}$  \cite{20}, now the solutions are even  more general  than Einstein's metrics \cite{21}. In this respect,  the TMG theory is  attractive because it is close  to real gravity.

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Dan – Drew Brees at his current age, I think just narrowly misses out on a top 5 spot, behind Brady, Rodgers, Carr, Wilson and Luck. There is however a possibility that he might be top 5 all-time, but that's for another time.
John – What more can you say about the man that hasn't already been said before he is a Hall of Famer has records better than anyone and is I mind one of the best to ever play the game.
Dave M – True. Absolutely he is. He has been doing this at too high a standard for too long not to be. Who've you got ahead of him? Wilson? Give me a break. Luck? The guy can barely stay on the field! Ryan? Last season is the outlier in his career, not the norm. Drew Brees is one of the greatest ever and a sure fire Hall of Famer.
Dave G – False. Brady, Rodgers, Luck, Wilson, Newton and perhaps as many as three others are ahead of him now for me. He is always going to put up stats, and he is still great, but if I am drafting for one season right now, he isn't in my top five.
John – It's going to be a tight affair between these two both love to put up points. I feel that the Saints will put up a big score that'll be just to much for the Dolphins.
Dave G – False. I just think Miami have a bit more about them. However, they will need to show way more on offence than they did against the Jets.
Dave M – False. I don't know. Last week the exact opposite happened to what I predicted and this one has a similar feel, except that the teams are closer matched.. Brees has often struggled on the road and on grass and Ajayi is going to have a monster game, so my vote goes to Miami.
John – It could be we are getting to see two Hall of famers this week in Brees and Peterson . But I think that the Rams Vs Cards will be the best game as it's going to be tight and it's a divisional game.
Dan – True! I certainly think this will be the closest. That being said it depends a lot on what Jay Cutler we get. If we get the player that lives up to his physical talents this should be a fun shoot out.
Dave M – False. The Cards/Rams game will be the closest. This one has a feel of last weeks, I'm not sure which team will win, but I think the most likely result is a blowout.
Dave G – True. It certainly seems to be the most intriguinh.
Jay Ajayi will have more than 100 yards rushing.
Dan – True. This still isn't a great Saints defence. I think Ajayi can take full advantage of that.
Dave M – True. Sean Payton's entire game plan is simply to outscore you. It doesn't matter if Jay Ajayi has 150 yards. Brees, Thomas, Ingram, Kamara et al, these guys could put up 500 yards of total offence almost at will, they don't give a damn about defence.
Dave G – True. He really should do anyway. This is a poor Saints defence and he should have a big day.
John – He only rushed for 16yards last week this week I think be is really going to break out a few times I'm going 150 and 2 TD.


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text:
Története
Az iskola történetének kezdetei 1886-ra nyúlnak vissza, amikor az Országos Tejgazdasági Felügyelőség célul tűzte ki maga elé, hogy minél előbb megszervezi hazánkban a szakoktatást. Az állam a legalkalmasabbnak látta egy olyan helyen létesíteni a szakiskolát, ahol rendelkezésre állnak a tej termeléséhez és feldolgozásához szükséges épületek és berendezések. Így esett a választás Lajos bajor királyi főherceg és Estei Mária Terézia Dorottya főhercegnő Vas vármegyei birtokára, ahol 1889-ben megalapították, majd 1890. augusztus 15-én megnyitották a Magyar Királyi Tejgazdasági Szakiskolát, az ország első ilyen jellegű oktatási intézményét.

A tanintézet alapításának célja a Szervezeti Szabályzatban olvasható:

A szerződés lejárta után - mely a herceg és a Földművelésügyi Minisztérium között köttetett a láncpusztai major szakoktatási célokra történő kézbeadásáról - a szakiskola 1929. november 24-én átköltözött a Sopron vármegyei Csermajorba, herceg Esterházy Miklós uradalmába, ahol 1920 óta már működött a herceg Esterházy Miklós-féle M. Királyi Mezőgazdasági Szakiskola.

Az I. világháború után Magyarország vezetése mindent megtett annak érdekében, hogy az emberekkel feledtesse a háború borzalmait. A magyar arisztokrácia több olyan alapítványt hozott létre, amelyek nemcsak a maguk dicsőségét, hanem az egész nemzet érdekeit szolgálták.

Ilyen alapítvány volt az is, melyet az ország leggazdagabb hitbizományosa, Galántai és Fraknói Esterházy Miklós herceg tett egy mezőgazdasági iskola létesítésére."Érezte és tudta azt, hogy Szent István koronája csak akkor kaphatja vissza régi fényét, ha ismét fel tudja építeni mezőgazdasági kultúráját. Az iskola alapításával biztos és határozott utat mutatott a nemzetnek a feltámadásra."Az adományozó Esterházy Miklós herceg nem élhette meg.az alapítványában megfogalmazott célok megvalósítását.  Fia, herceg Esterházy Pál teljesíteni kívánta apja elhatározását, így az alapítványban felajánlott 200 kh ingatlant tartozékaival együtt 1920. október 19-én átadta az Államkincstárnak

A Csermajori Tejipari Szakiskola 1930-as években végzett szakemberei, sajt- és vajmesterei egyedül adták a hazai tejipar meghatározó szakembergárdáját. Ők álltak a kádak és köpülők mellett, szereztek hírnevet a magyar sajtnak és a többi tejterméknek, szervezték újjá a II. világháború után a magyar tejipart.

Döntő jelentősége volt a tejgazdaságok és tejüzemek megszervezésénél, a hazai technológiák kidolgozásánál és elterjesztésénél a magyaróvári Tejgazdasági Kísérleti Intézetnek, amelyet 1903-ban alapítottak Magyar Királyi Tejkísérleti Állomás néven. E téren úttörő munkát végzett az akadémia több kiváló professzora: Ujhelyi Imre, Grátz Ottó, Budai Gyula, Vas Károly, Vladár Endre, Bánvárth Sándor, Ejury Lajos, Csiszár József és még sok más kutató. A csermajori tanműhely gyakran szolgált a sajtgyártási kísérletek színhelyéül, vagyis számos hazai sajt technológiájának kidolgozásánál meghatározó szerepet játszott a Csermajorban felhalmozott szakmai tapasztalat.

A nagy álom, a nagy terv megvalósítói, a "magyar mester, magyar sajt" elv diadalra vitelében a manuális munka zömét a Csermajorban tanult szakemberek segítségével végezték el!

Mezőgazdasági és tejipari szakképzés Csermajorban

Az intézményben 1929-től 3 szervezeti egység működött közös igazgatás alatt:

 Tangazdaság
 M. Kir. Mezőgazdasági Szakiskola
 M. Kir. Mezőgazdasági Tejipari Szakiskola

A csermajori tangazdaság alapját az az alapítványi birtok adta, melyet herceg Esterházy Miklós létesített és fia, Esterházy Pál valósított meg. A 200 katasztrális holdat kitevő alapítvány az Országos Földbirtokrendező Bíróság 1923. január 25-én kelt ítéletével 376 kh 1545 négyszögöl területtel, 1930-ban pedig 3 kh 1454 négyszögöllel bővült.

A tangazdaságra épült Csermajor mindkét iskolája, melyek működéséhez ez a gazdaság teremtette meg a szükséges anyagi feltételeket, és adta meg tanulóinak a gyakorlati tudás megszerzésének lehetőségét is. A tangazdaság rendelkezett mindazokkal a felszerelésekkel, illetve szakszerű vezetéssel, amelyekre és akikre az irányításhoz szükség volt. Ennek hatása már a 20-as, de különösen a 30-as években is meglátszott, mert a gazdaság igen jó eredményekkel büszkélkedhetett. A mezőgazdasági szakiskola, később a tejipari szakiskola létesítésével a gazdaságban az állandó alkalmazottak mellett az iskolák tanulói is dolgoztak, így sajátítva el a későbbi önálló munkavégzéshez elengedhetetlen ismereteket és a rutint. Kivették részüket a földművelésből: kaszáltak, kapáltak, illetve az állatokat is gondozták: fejték, takarmányozták és ápolták a teheneket, juhokat, lovakat, sertéseket és háziszárnyasokat. A tangazdaság minden részterületén szigorú rendben folyt a munka, amihez a tanulóknak is alkalmazkodniuk kellett.

A M. Kir. Mezőgazdasági Tejipari Szakiskola célja, hogy a tanulók a tej termelése, kezelése, feldolgozása és értékesítése kapcsán felmerülő valamennyi munkafázist elsajátítsák és "kellő kézügyességre tegyenek szert" tanterve különbséget tett a gyakorlati és az iskolai évfolyam között. A tanév során a legnagyobb hangsúlyt a gyakorlati oktatásra helyezték. A tanulók kötelesek voltak minden munkában részt venni, valamint az előírt hetességi szolgálatot ellátni, mely során az alábbi feladatokat végezték:

 kazán- és gőzgépkezelés,
 fölözés,
 vajgyártás,
 sajtgyártás,
 sajtkezelés,
 tejvizsgálat,
 fejés,
 szobarend,
irodai munka.

A Csermajori herceg Esterházy Miklós-féle M. Kir. Mezőgazdasági Szakiskola 1920 őszén kezdte meg az oktatást. Az iskola célját az igazgatóság az alábbi módon határozta meg."A Mezőgazdasági Szakiskolánknak célja az, hogy jól képzett, a saját lábán megállni tudó, magyar nemzeti szellemtől áthatott gazdaifjakat neveljen. A saját gazdaságába visszatérő ifjú gazda községében előharcosa legyen a mezőgazdasági kultúrának és példaadásával megfizethetetlen szolgálatot tegyen az egész községnek."Az egyes tantárgyak anyagát a fenti célok megvalósításának megfelelően állították össze, különös tekintettel a gyakorlati ismeretekre, melyek elsajátítása csak úgy volt lehetséges, ha a gazdasági munkákat meghatározott időn keresztül a tanulók végezték. Tanári felügyelet mellett minden tavaszi, nyári, őszi növény- és állattenyésztési munkában részt vettek. A gazdaság vezetését, a különféle munkák irányítását a hetesi szolgálat alatt sajátították el. A hetesek önállóan, felelősséggel intézkedtek, de ha szükség volt rá, a vezető tanároktól és a majorgazdától minden segítséget megkaptak.

A téli időszakban a tanulóknak csak az állatok etetése, gondozása és fejése adott munkát, így jutott idő a különféle háziipari foglalkozásokra is. Megismerkedhettek a kosárfonással, kefekötéssel és seprűkészítéssel. Ezeket később jól hasznosíthatták. Az ügyesebbek a szerszámok javításakor megtanulták az újak készítését is.

A tantestület több tagja mezőgazdasági szakcikkeivel is hozzájárult a korszerű növény- és állattenyésztési módszerek elterjedéséhez. 1933-ban baromfitenyésztő-telepet létesítettek, ugyanakkor az iskola tehenészetével belépett a Sopron Vármegyei Szarvasmarha-tenyésztő Egyesületbe. Ez utóbbi nagy elismerése volt az oktatási intézményben folyó szakmai-gyakorlati munkának.

Szakképzés a II. világháború után
Az 1945/46-os tanévtől kezdődően a szakiskola feladatköre kibővült. Az intézmény neve az új tartalomnak megfelelően:  Csermajori Állami Mezőgazdasági Tejipari Szakiskola és Tejgazdasági Szaktanácsadó Állomásra módosult.

Csermajor az 1946/47-es tanévig kettős intézményként - tejipari szakiskola és gazdaképző iskola - működött, de a következő tanévtől a mezőgazdasági iskolát megszüntették, mert az akkor létrehozott Kapuvári Gazdaképző Iskolába költöztették át.

1949-ben is nagy változás történt: Mezőgazdasági Gimnázium Tejipari Tagozataként átalakult az oktatás anyaga. A szakmai tárgyak mellett jelentős mértékben megnőtt a közismereti tantárgyak óraszáma.

Technikusképzés
Az 1950-es esztendő két okból is fontos dátum Csermajor életében: megindult a technikumi képzés, másrészt a mezőgazdasági szakképzés megszűnése miatt a tangazdaság átadásra került a Pusztacsaládi Állami Gazdaság számára. Ez kb. 400 kh területet, istállókat, állatállományt és felszereléseket jelentett. Csak a tejipari oktatási célokat közvetlenül szolgáló létesítmények maradhattak az iskola birtokában. Az átadás egyszerű könyvjóváírással történt!

Az iskola felettes hatóságai (az Oktatási, valamint az Élelmezésügyi Minisztérium) az oktatás időtartamát 1954-től 4 évre emelték és a végzett hallgatók az érettségivel egyenértékű tejipari technikusi oklevelet kaptak.

Az iskola fenntartója 1950-től az Élelmezésügyi Minisztérium lett, amely igen sokat tett az oktatás személyi és anyagi feltételeinek biztosításáért. Korszerűen felszerelt mikrobiológiai és kémiai laboratóriumot bocsátott a technikum rendelkezésére

Szakközépiskola

Az 1964/65-ös tanév ismét változásokat hozott iskolánk életében: felmenő rendszerben elkezdődött a szakközépiskolai oktatás. Az első osztályosok már új tanterv és óraterv szerint haladtak. Más lett a képzési cél, és más lett a 4 év után kapott bizonyítvány és képesítés is. A végzős tanulók a IV. osztály befejezése után érettségi- és szakmunkásvizsgát tettek, így érettségizett tejtermékgyártó szakmunkásokká váltak.

Az élelmiszeripari szakközépiskolában a műszaki tárgyak nagyobb súlyt kaptak. Megjelent a géprajz, műszerismeret, az élelmiszeripari műveletek és folyamatok és géptani gyakorlat. Sajnos csökkent a laborgyakorlatok óraszáma és megszűnt a mikrobiológia, valamint a szakkémia.

Az 1967/68-as tanévtől kezdve a 4 éves szakközépiskolai oktatás mellett megindult a 3 éves szakmunkásképzés, melynek lezárásaként a tanulók szakmunkásvizsga után tejtermékgyártó szakmunkás képesítést szereztek.

A tejtermékgyártó szakmunkásképzésben dunántúli központtá vált intézményünk. 1967-től 1976-ig Székesfehérváron, 1971-től 1976-ig Zalaegerszegen működött kihelyezett osztályunk. Az 1970-es évek közepétől Csermajor döntő szerepet játszott előbb a Dunántúlon szervezett vállalati felnőtt szakmunkásképző tanfolyamok, majd a technikusminősítő képzések és vizsgáztatás lebonyolításában is.

1968 mérföldkő az iskola történetében, hiszen felvette a tejgazdaságtan egyik jeles hazai oktatója, Ujhelyi Imre nevét, innentől Ujhelyi Imre Élelmiszeripari Szakközépiskola és Szakmunkásképző Intézet néven működött.

1989-től a képzési idő 4-ről 5 évre emelkedett. A tanulók az első négy évben a szakmai tárgyak mellett (tejipari technológia, szakmai gépek stb.) jelentős óraszámban közismereti tantárgyakat (matematika, magyar nyelv és irodalom stb.) is tanultak. A 4. év végén érettségi vizsgát tettek, majd az 5. évben csak szakmai tárgyakat tanulva felkészültek a technikusi vizsgára. Így a tanév végén a sikeres vizsga után tejipari technikusokká váltak.

1996-ban új képzési formaként beindították a szakmunkások szakközépiskoláját, ahol a szakmunkás bizonyítvány birtokában, két év alatt érettségit és számítástechnikai végzettséget szerezhettek a tanulók.

1998-tól az eddigi hagyományos képzési rendszer ismét megváltozott.

A 9-10. osztályban csak általános élelmiszeripari ismereteket tanítottak, és csak a 10. osztály végén kellett szakirányt választani, vagyis a tanulóknak csak ekkor kellett eldönteniük (tehát, nem közvetlenül a 8. osztály elvégzése után), hogy az általános élelmiszeripari szakképzés keretében tanulnak-e tovább vagy inkább a tejtermékgyártó szakmunkás, illetve a falusi vendéglátó szakképesítés megszerzését választják.

Szakképzés felsőfokon
Kiemelkedő jelentőségű dátum iskola történetében 1999 szeptembere, amikor az Élelmiszeripari Egyetem Szegedi Főiskolai Karával közösen indíthatta Csermajor a felsőfokú Élelmiszeripari menedzser szakképzési formát, amelyben érettségizett fiatalok vehettek részt, 2 éves tanulmányi idővel.

A képzési kínálat bővítésével egyidejűleg az iskola neve is módosult: Ujhelyi Imre Élelmiszer-ipari Közép- és Felsőfokú Szakképző Iskola és Kollégium lett.

Tanüzem

A csermajori iskola tanüzemét 1929-ben létesítették 968 m² alapterületen, melyben sajt- és vajműhelyt, laboratóriumot és érlelőpincéket alakítottak ki. Kezdetben működése manufaktúra jellegű volt: a sajtokat jobbára fából készült berendezésekben, illetve fémüstökben készítették. A korszerűtlen technikai adottságok ellenére az ott dolgozó mesterek kiváló minőségű sajtokat és vajat gyártottak.

Az 1930-as években tanüzem felszereltsége színvonalában már felülmúlta a magyar tejipar műszaki-technológiai átlagát.

A tanüzem a tejipari gyakorlatok képzési helye volt, amelyet többször korszerűsítettek, fejlesztettek A tanulók szakoktatói irányítás mellett önállóan végezték a tej feldolgozását, a különféle tejtermékek gyártását. A Csermajori tanüzem az országban még mindig egyedülállóan jó feltételeket teremt a tejipari szakképzéshez.

A tanüzemben gyártott tejtermékek: vaj, tejföl, túró, valamint a tanüzem pincéjében érlelt sajtok. A vásárlók körében a legkedveltebb csermajori sajtok a következők voltak: Ilmici (Mosoni megyei csemege), Pálpusztai, Teasajt, Lajta és a Trappista.

Iskolaösszevonás
2009-ben a közös fenntartó a csermajori iskolát összevonta a fertődi Porpáczy Aladár Kertészeti Szakközépiskolával, és a két iskolából létrehozták a Győr-Moson-Sopron Megyei Önkormányzat Általános Művelődési Központját, aminek egyik tagiskolája lett Csermajor.

Ezáltal 2009. augusztus 1-jével a csermajori iskola önállósága – alapításának 120. évében – megszűnt.

Az összevonás világos és egyértelmű célja az volt, hogy az új intézményben a csermajori és a fertődi iskola két egyenrangú intézményként egymást erősítve a régió egyik legmeghatározóbb szakképző iskolája legyen. Ennek ellenére már 2014-ben, öt évvel az összevonás után nem volt egyetlen csermajori tanuló sem a "közös" intézményben, a csermajori tanárokat és dolgozókat az összevonást követő néhány év alatt szétszélesztették. Csermajorban megszüntették a tanítást, és emiatt a tejesek Sárospatakja ma már nem lehet része a magyar közoktatásnak és szakmai képzésnek, eltűnik egy hungarikum.

Sajtverseny Csermajorban

Hazánkban a tejtermékek minőségének megőrzésére és javítására való törekvés miatt a 83.700/1928. számú FM rendelet alapján hozták létre a Tejtermékek Magyar Királyi Ellenőrző Állomását. Az intézmény alapításának 70. évfordulójára emlékezve kereste meg a csermajori iskola a sajtgyártó üzemeket, cégeket, és hívta a szakma képviselőit egy országos találkozóra, termékminősítésre. Mivel a felhívás kedvező fogadtatásra talált, ezért az iskola 1998. december 3-án megrendezhette az első Csermajori Országos Sajtversenyt. Az egyre növekvő érdeklődés mellett Csermajorban tizenötször hirdették meg a sajtkészítők e rangos találkozóját, alkalmat adva számukra, hogy termékeiket bemutathassák a szakmának és a sajtkedvelő közönségnek.

Az utolsó sajtversenyt, az iskolaösszevonás után 2012-ben rendezték meg, vagyis miután megszüntették Csermajorban a tanítást, megszűnt a csermajori sajtverseny is.

Források

Pappné Szabó Alexandra: A csermajori középiskola története (1889-2002) (Zsámbék : s.n., 2002)
Németh Imre: Csermajor (Kapuvár : s.n., 1976)
Somogyi Imre: Láncpuszta - Csermajor, 1889-1989 : centenáriumi évkönyv (Csermajor : Ujhelyi Imre Élelmiszeripari Szakközépiskola és Szakmunkásképző Intézet, 1989)
Pappné Szabó Alexandra: A csermajori középiskola könyvtárának rövid története (Könyvtári Levelező/Lap. - 15. (2003))
Sajtverseny Csermajorban -2008 (Turizmus Online)
Győr-Moson-Sopron Megyei Értéktár - 2016
 Papp László: Csermajor anno (A csermajori középiskola honlapja)

redpajama_set_name: RedPajamaWikipedia


I went out to take a picture of the moon with my new camera (nikon coolpix b500) but there was no moon out, and then i saw a very dim white light just over the clouds, and then another one appeared next to it, so i started taking pictures, and the camera was on (moonlight setting) so it saw stuff that i couldn't see with the naked eye, such as another white light that completed a perfect triangle formation in the clouds, along with a transparent green/blue flying saucer with lights on the inside, flying under the clouds, just over my head. then the green/blue craft flew up in the clouds, then emitted some sort of green aura. but what is insane is that about 5 hours later at my cousins house about 2-3 miles away from my house, we all saw the giant lights in the clouds, again. so i took more pictures, and then it started to form the same triangle, next thing i know, it was right over me again.... yep, twice in one night, and i got the photos to prove it. the first sighting happened for about 30 minutes. and the second sighting only happened for about 7 minutes.


Ground source systems offer homeowners heating and cooling at a fraction of the cost of traditional HVAC. According to the Environmental Protection Agency (EPA), you'll save 30 to 70 percent on heating costs and 20 to 50 percent on cooling costs. A typical system has an efficiency rating of 400 percent, whereas a high-efficiency gas furnace has a rating of 94 percent.
These systems can give you hot water too. In summer, they actually use the warmth from your home to heat water. They can also make ice, melt snow and heat your pool.
The average indoor unit of a ground source heat pump lasts for over 20 years. The underground system of pipes outside can last for up to 50 years. HVAC maintenance is minimal.
According to the EPA, ground source heat pumps are the most cost-effective and environmentally-safe systems available today. The below-ground, solar-generated heat they use is a free energy source that will never run out. These systems are clean, safe, and generate no pollution.
When the initial cost of a ground source heat pump is covered by low-interest financing, the savings on utility bills will cover monthly repayments with cash left over. Most systems pay for themselves in three to five years.
To learn whether Trane geothermal technology would be a good fit for your Denver, NC, home, or to schedule a free home heating and cooling consultation, visit Lakeside Heating & Air Conditioning or call 704-208-4172 .


Just like less is more, bigger isn't always better – especially when referring to a retailer's store size.
Retailers across the country are rightsizing their brick and mortar stores to attract new customers and adapt to a changing retail environment that's being driven by the need for omnichannel strategies and the desire to expand into urban environments – among other factors.
For these retailers, recalibrating the size of their stores means less square footage, and that requires retailers and developers to re-evaluate their development and expansion strategies.
Because even though they're moving to smaller footprint stores, they're most likely going to be faced with a higher rent than what they were being charged in their larger box spaces.
Not to mention the even more difficult challenges of determining the right types and right amount of merchandise to fit into the limited space available while still telling their brand story, sharing their brand values and connecting with their current and potential customers.
Those retailers who are rightsizing their stores have to develop a strategy that determines what elements of their overall brand to import into the smaller store design and experience in order to execute successfully in the new space.
Given that, there are no one-size-fits-all strategies. It's important for you to remember that the assumptions that may hold true for other property types don't always work for smaller locations or urban locations. There are too many other factors and variables at play.
Luckily for retailers, more advanced technological and analytical tools have been developed to help assist retailers with making the best decisions when it comes to picking the right store location, determining the right size store, and forecasting how each size store and market will affect their bottom line.
Rightsizing your store network is a potentially perilous undertaking. That's why it's still imperative that retailers pick the best markets and the best sites within those markets for their new locations, determine the optimal number of locations for each market and keep up with consumer expectations of what the in-store experience should be.
The good news is that with customer data and insight, retailers can understand their customers, understand how their best customers want to interact with their brand and deliver an experience that meets their customers' expectations.
Analytic insights help retailers keep their best customers' desires and needs in focus when assessing the potential of new markets and new sites.