Why is symmetry so important in particle physics?
I find it incredibly frustrating that again and again, the Lagrangian gets introduced and then said it should be minimized without ever explaining the motivation behind doing so. What <i>is</i> the Lagrangian and <i>why</i> should it be minimized?<p>I totally get how L gets defined mathematically, how it is derived from Newton&#x27;s laws (this part is typically well explained by textbooks), and why in the case of point particles, a curve that violates Newton&#x27;s laws does not minimize L. But there is no understanding at all, just saying &quot;okay, it checks out&quot; on a math level.<p>It doesn&#x27;t help at all that on a math level, L isn&#x27;t actually minimized but its derivative set to 0, which isn&#x27;t even equivalent to &quot;minimized or maximized&quot;. Why doesn&#x27;t a single textbook explain why maximizing L is also okay when they first stated &quot;minimized&quot;? Or why derivative=0 is sufficient? As a reader, I always get the impression that &quot;well, of course they cannot explain that, because they don&#x27;t even know why L should be minimized in the first place&quot;. It&#x27;s just all formulas that are easy to verify but don&#x27;t convey a single bit of understanding.<p>Just for comparison, I found quantum mechanics based on the Schrödinger equation and the Hamiltonian rather easy to grasp, because every piece of it has an easy-to-understand meaning, that also gets explained really well. Why is this seemingly impossible for the Lagrangian?