Calculus of Variations. Lagrangian Hamiltonian Mechanics Mathpages. Over at http://www.mathpages.com/home/kmath523/kmath523.htm
is an article about Lagrangian and Hamiltonian Mechanics
with a derivation of the Euler-Lagrange equations of motion.
Mid-way through is this statement:
"Variations in x,y,z and X at constant t are independent of t (since each of these variables is strictly a function of t), so we have"
$ \frac{\partial x} {\partial X} = \frac {\partial \dot x} {\partial \dot X} $ ;
$ \frac{\partial y} {\partial X} = \frac {\partial \dot y} {\partial \dot X} $ ;
$ \frac{\partial z} {\partial X} = \frac {\partial \dot z} {\partial \dot X} $
Can someone please explain why this is true.
To me it seems wrong because by applying the chain rule I get:
$ \frac {\partial \dot x} { \partial \dot X} = \frac {\partial (\frac {dx} {dt})} {\partial (\frac {dX} {dt})} $
$ = \frac {\partial (\frac {\partial x} {\partial X}\frac {dX} {dt} + \frac {\partial x} {\partial Y}\frac {dY} {dt}) } {\partial (\frac {dX} {dt})} $
$ = \frac {\partial (\frac {\partial x}{\partial X}\frac{dX}{dt})} { \partial (\frac {dX} {dt})} + \frac {\partial (\frac {\partial x} {\partial Y}\frac {dY} {dt}) } {\partial (\frac {dX} {dt})} $
$ = \frac {\partial x} { \partial X} + \frac {\frac {\partial x} {\partial Y} \dot Y } {\dot X} $
$ \ne \frac {\partial x} { \partial X} $
Thanks

It is confusing because there is some serious abuse of notation going on in classical mechanics and it results in unclarity when taking partial derivatives. Let's see what's going on: 
We have two coordinate systems $(x,y), (X,Y)$ related by a transformation $(x,y)=(x(X,Y),y(X,Y))$ and we have a curve $\displaystyle (X,Y)=(X(t),Y(t))$ (see I have already abused notation), so we have  a curve $\displaystyle (x,y)=(x(X(t),Y(t)),y(X(t),Y(t)))\,.$  Using chain rule we get $\displaystyle \dot{x}=\frac{\partial x}{\partial X}\dot{X}+\frac{\partial x}{\partial Y}\dot{Y}.$ There has been some abuse of notation so far, but I can make sense of everything, and in fact this notation is unfortunately standard. Now doing the obvious thing gives you what you want.
But the last part is one which I find confusing; what does $\displaystyle \frac{\partial \dot{x}}{\partial\dot{X}}$ really mean? Is it a directional derivative of some sort? I think this part was never really clear to me.
