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Define the energy function $\mathcal{E}={\sum }_i{\dot{q}}_i{p}_i-L$. Along solutions of the Euler-Lagrange equations,ConsequentlyFor an ordinary isolated mechanical system with $L=T-U$, a time-independent quadratic kinetic energy and a velocity-independent potential energy, ${\sum }_i{\dot{q}}_i{p}_i=2T$, so $\mathcal{E}=T+U$. For a more general Lagrangian, the conserved quantity is the displayed energy function; identifying it with $T+U$ requires these assumptions.

Generalized coordinates are variables $q_1,\dots ,q_n$ that specify the allowed instantaneous configuration of a mechanical system. They need not be Cartesian positions: a rigid pendulum's angle is a suitable single coordinate. With constraints solved, the particle positions are functions ${\mathbf{r}}_i(q_1,\dots ,q_n,t)$.