Let $M$ be a manifold and fix a Lagrangian $L\in C^\infty(T M )$. Let $x\_1,\dots x\_n$ be local coordinates for $M$ and equip the tangent bundle and cotangent bundle with standard coordinates $(x\_1,\dots, x\_n, v\_1,\dots, v\_n)$ and $(x\_1,\dots, x\_n,\xi\_1,\dots, \xi\_n)$ respectively. The Legendre transform associated to $L$ is $$\Phi\_L: T M\to T^\ast M \ , \ (p,X\_p)\mapsto \Phi\_{L\_p}(p,X\_p):=(p,\frac{\partial L}{\partial v}(p,X\_p))$$We also define the dual function associated to $L$ which is $$L^\ast:T^\ast M\to\mathbb{R} \, \ (p,\alpha\_p)\mapsto \min \left\{\alpha\_p(X\_p)-L(p,X\_p) \ , \ (p,X\_p)\in T M\right\} $$ Under certain conditions on $L$, we get that $\Phi\_L$ is a diffeomorphism with inverse $$\Phi\_L^{-1}(p,\alpha\_p) = (p,X\_p) \text{ where $(p,X\_p)$ minimizes } \alpha\_p-L$$It can be shown that $$\Phi\_L^{-1}=\Phi\_{L^\ast}$$My question is about the following claim.
$$\text{If } \Phi\_L(p,X\_p)=(p,\alpha\_p)\text{ then } \frac{\partial L}{\partial x}(p,X\_p)=-\frac{\partial L^\ast}{\partial x}(p,\alpha\_p)$$ This is proved in Da Silva's book "Lectures on Symplectic Geometry" in the chapter on Hamiltonian mechanics. I do not understand her proof. Letting $H=L^\ast$, the definition of $H$ is $H(x\_1,\dots x\_n,\xi\_1,\dots,\xi\_n)=H(x,\xi)=\xi(x,W)-L(x,W)$ where $(x,W)=\Phi\_L^{-1}(x,\xi)=\Phi\_H(x,\xi)$ or equivalently $\xi=\Phi\_L(W)$ so that $\xi$ is a function of the $x\_i's$ and $v\_i's$. She takes the total derivative of both sides of this equation with respect to $x$ to get $$\frac{\partial H}{\partial x}+\frac{\partial H}{\partial \xi}\frac{\partial \xi}{\partial x}=\frac{\partial\xi}{\partial x}v-\frac{\partial L}{\partial x}$$ The left hand side is just the chain rule, right? But how is the right hand side obtained? If $\xi$ is dependent on $x$ and $v$ then isn't $v$ dependent on $x$ and $\xi$? I believe that once we have this formula the hypothesis gives us the result as $\frac{\partial H}{\partial \xi\_i}(p,\alpha\_p)=v^i(p,W\_p)$. If someone could walk me through this calculation it would be greatly appreciated, I have spent more time on this then I would like to admit.



Computation with the Legendre Transform
I actually was interpreting what she was saying wrong. The coordinates $\xi$ on $T^\ast M$ are of the form $\xi=\Phi\_L(x,v)$ so that $\xi\_i=\frac{\partial L}{\partial v\_i}$ and the Legendre transform is involutive so $\Phi\_L^{-1}(\xi)=\Phi\_{L^\ast}(\xi)=v$. This makes the computation fairly straightforward.


