Lagrangian energy function ($\mathcal{E}={\sum }_i{\dot{q}}_i\partial L/\partial {\dot{q}}_i-L$)

For a Lagrangian $L(q,\dot{q},t)$, its energy function satisfies $d\mathcal{E}/dt=-\partial L/\partial t$ along the Euler-Lagrange equations. This follows by differentiating and cancelling the ${\ddot{q}}_i\partial L/\partial {\dot{q}}_i$ terms. It is conserved when $L$ has no explicit time dependence. For imposed moving constraints it need not coincide with inertial mechanical energy; a Jacobi integral is an important example.