# If velocity is a function of position and position is function of time, is velocity a function of time as well?

I was reading through Lagrangian Mechanics and noticed this sort of question popping up everywhere. To illustrate, the 3d x coordinate was considered to be a function of X, Y (degrees of freedom). Now in the differentiating mess, the authors considered both x and X as functions of time. But I thought x was a function of X and hence the confusion.

• Yes, velocity can be considered as a function of position rather than as a function of time. – G. Smith Jul 4 '19 at 17:37
• @G.Smith Stated this way I am afraid that there could be some problem with counting the number of independent degrees of freedom. – GiorgioP Jul 4 '19 at 17:53
• I don't know if this is what you're getting at, but velocity is not in general a function of position, because an object can have two different velocities at the same position, if it returns to the same point at two times. – user4552 Jul 4 '19 at 17:58
• @ Ben Crowell Thanks for making that point. Lemme edit the question – Eesh Starryn Jul 4 '19 at 18:01
• Maybe you could show the math you are talking about? – user234190 Jul 4 '19 at 19:34

If Lagrangian is written as $$L=L(x(t),v(t),t)$$, $$L$$ is time-dependent explicity through $$t$$ and implicitely through $$x$$ and $$v$$. If $$L=L(x(t),v(t))$$, then $$L$$ is not explicitly dependent on time but implicitly dependent on time through $$x$$ and $$v$$. This means $$L$$ can change if and only if $$x$$ and $$v$$ changes with time and can't change with time without a change in $$x$$ and $$v$$.

Same explanation goes for all the functions depending explicitly and implicitly on time.