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If my thinking is wrong please let me know. I have little knowledge on beyond-college physics. For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,...

Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$ (i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q_{i},\dot{q}_{i},t)$, $i=1,2,......