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I am studying Symplectic Geometry and I was wondering how one can compute Euler-Lagrange equation in a coordinate free manner. For instance, I know for the following Lagrangian $L(x,v)=\frac{1}{2}g_{x}(v,v)-V(x)$ the corresponding equation is $\nabla_{\dot\gamma}\dot\gamma=-\nabla V$, but it is a bit complicated to derive it in a coordinate system.

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  • $\begingroup$ This is a topic in Hamiltonian dynamics, with canonical variables. Goldstein or Landau has a good discussion of it. $\endgroup$ Feb 12, 2020 at 21:24

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