Does $L=T-V$ still hold when $L$ is NOT time-dependent?

Question

I am aware that the Lagrangian $L=T-V$ where $T$ is the kinetic energy and $V$ is the potential energy when $L$ depends on, for example, $r, \dot{r}, t$. My question is, does this still hold when the Lagrangian does not explicitly depend on time, i.e. I have $L(r,\dot{r})$ instead?

Answer 1

Of course. Indeed most Lagrangians do not depend explicitly on time, v.g. for the harmonic oscillator $$ L(x,\dot{x})=\frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 $$ doesn't depend explicitly on $t$.