Noether conserved quantity for a mechanical point symmetry ($J={\sum }_i{p}_i{\xi }_i$)

If a time-independent Lagrangian $L(q,\dot{q})$ is invariant under a one-parameter point transformation with infinitesimal generator ${\xi }_i={\partial }_s{Q}_i{|}_{s=0}$, then every Euler-Lagrange trajectory conserves

An infinitesimal transformation that preserves the action up to endpoint terms. For $\delta t=\epsilon \varphi$ and $\delta q_i=\epsilon {\psi }_i$, invariance of $Ldt$ yields the Noether charge ${\sum }_i{p}_i{\psi }_i-H\varphi$, less any boundary-term generator.