Is there a physical connection between Lagrangian mechanics and Hamilton mechanics

Question

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 $L$?