# In proving that the Lagrange equations hold in any coordinate system

1. Oct 10, 2013

### M. next

I was checking the proof of this, when things came vague at one point.

It goes as follows, how to prove that Lagrange's equations hold in any coordinate system?

Let q$_{a}$ = q$_{a}$(x$_{1}$,..., x$_{3N}$, t)

here the possibility of using a coordinate system that changes with time is included.

By chain rule: $\dot{q_{a}}$= $\partial$q$_{a}$/$\partial$x$^{A}$($\dot{x^{A}}$) + $\partial$q$_{a}$/$\partial$t

To be in a good coordinate system, we should be able to invert this relationship, call this equation (1):

$\dot{x_{a}}$= $\partial$x$_{a}$/$\partial$q$^{A}$($\dot{q^{A}}$) + $\partial$x$_{a}$/$\partial$t

Then the book said, now we can examine L(x$^{A}$, $\dot{x^{A}}$) when we substitute in x$^{A}$(q$_{a}$, t)

Using (1), we have

$\partial$L/$\partial$q$_{a}$ =

($\partial$L/$\partial$x$^{A}$)($\partial$x$^{A}$/$\partial$q$_{a}$) + $\partial$L/$\partial$$\dot{x^{A}}$($\partial$$^{2}$x$^{A}$/$\partial$q$_{a}$q$_{b}$*$\dot{q_{b}}$ + $\partial$$^{2}$x$^{A}$/$\partial$q$_{t}$q$_{a}$)

My question is I don't understand where the q$_{b}$ come from??