I know that the Hamilton's equations results from the Legendre transformation of the Lagrange equation. It's also well-known that the Hamiltonian is equal to the energy of a system if the system is time independent but that is caused by the symmetry of $H$, not because of the fact that the Lagrange function is given by the kinetic energy plus the potential energy. So is there a concret physical relationship between $H$ and $L$?
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$\begingroup$ Possible duplicates: physics.stackexchange.com/q/4384/2451 , physics.stackexchange.com/q/105912/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Mar 20, 2019 at 14:53
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