Lagrangian & conservation of energy

Question

I'm working through Garrity's "E & M for Mathematicians." An exercise is to show that if the Lagrangian $L = T - U$, where $T$ is a function of $x'$, $y'$ and $z'$, and $U$ a function of $x$, $y$, and $z$ (i.e., $T$ and $U$ both not explicit functions of $t$), then total energy $T+U$ is a constant (his Corollary 6.4.1). But this seems trivial, because since both $T$ and $U$ are not functions of $t$, $T+U$ has to be constant (with respect to $t$), i.e., $d(T+U)/dt = 0.$ What am I missing?

Answer 1

The implicit time dependence in the position curve ${\bf r}: \mathbb{R}\to \mathbb{R}^3$ makes the statement of conserved total energy $T+U$ non-trivial, cf. comments by Paul and A.G.

Besides no explicit time dependence, it is assumed that the position curve $t\mapsto {\bf r}(t)$ satisfies the Euler-Lagrange (EL) equations. A position curve $t\mapsto {\bf r}(t)$ that does not satisfy the EL equations will typically not have constant total energy $T+U$.

By the way, this is an example of Noether's theorem.