I have worked with Thomas C.T. Michaels Analysis II to study the application of the implicit function theorem and I have a pretty solid idea of how to apply it to multivariate functions but I cannot solve the below problem:
Setup: Let $L \in C^2( \mathbb{R}^{2n+1}, \mathbb{R})$ be given by $L(t,x,v)=L(t,x_1, \dots , x_n, v_1, \dots , v_n)$ and consider a point $(t^*, x^*, v^*) \in \mathbb{R}^{2n+1}$ such that the Matrix $$\left( \frac{\partial^2 L}{\partial v_j \partial v_i}(t^*,x^*, v^*)\right)_{i,j=1, \dots ,n} \text{ is invertible} $$ Define $p^*:= \left(\frac{\partial L}{\partial v_1}, \dots , \frac{\partial L}{\partial v_n} \right)(t^*, x^*, v^*)$
Problem: Find an open neighborhood $U$ of $(t^*, x^*, p^*)$ and a function $\phi \in C^1(U, \mathbb{R}^n)$ such that for all $(t,x,p) \in U$ $$\frac{\partial L}{\partial v_i}(t,x,\varphi(t,x,p))=p_i, \forall i = 1, \dots , n $$
My approach: Besides the ugly formulation of the problem I don't think there is much to it rather than to cleverly apply the implicit Function Theorem but the abstraction of the problem makes me fail to go my naive route:
First I usually try to introduce a 'new' variable $z=\varphi(t,x,p)$, but I am at doubt if that is a clever step for this problem because I would want to verify that the point $(t^*, x^*, p^*)$ solves the equation: $$ \frac{\partial L}{\partial v_i}(t,x,\varphi(t,x,p))=p_i, \ \forall i =1, \dots , n \tag{*}$$ So $p$ seems to have two meanings in the above expression, one that is local for $\varphi$ the other that is global and asks for the $i-$th entry of $p$. Therefore I can only make sense out of (*) if I keep the unknown function inside and try to verify that $$\frac{\partial L}{\partial v_i}(t^*, x^*, \varphi(t^*,x^*, p^*))=p_i^*=\frac{\partial L}{\partial v_i}(t^*, x^*, v^*) $$ And although I know what $p^*$ and therefore $p_i^*$ look like, I cannot possible tell that the above equation is true or not.
Indeed if *(by some wizardry)*it happens to be that $$\varphi(t^*, x^*, p^*)=v^* $$ Then the equation is clearly solved, furthermore the matrix would be invertible and that would be a giant leap towards the solution of the exercise (if not entirely solve it), but nothing seems to hold together.
I also am confused by the notion of $\varphi$ as in $\varphi(t,x,p)$ lets say I have a function $f(x,y,z)$ and I want to solve it locally for the variable $z$ by the implicit function theorem, clearly I end up with a function $\xi$ that is dependent on $x,y$ namely $f(x,y,\xi(x,y))$ in the above question however there seems to sneak another variable in.