Let $Q$ be a smooth manifold, $\alpha\in\Omega^1(T^*Q)$ the tautological $1$-form and $F:T^*Q\to T^*Q$ a symplectomorphism such that $F^*\alpha=\alpha$ ($F$ could be the cotangent lift of some diffeomorphism $f:Q\to Q$, for example). Let $X\in\mathfrak{X}(T^*Q)$ be the Euler vector field, which is defined as $X:=\sum_{i=1}^n\xi_i\frac{\partial}{\partial \xi_i}$ in local coordinates $(x_1,…,x_n,\xi_1,…,\xi_n)$. Prove that $dF_m(X_m)=X_{F(m)}$.
I have an initial idea: $\alpha$ is given locally by $\alpha=\sum_{i=1}^n\xi_i dx_i$. If $F$ takes coordinates $(x_1,…,x_n,\xi_1,…,\xi_n)$ into coordinates $(x'_1,…,x'_n,\xi'_1,…,\xi'_n)$, the condition $F^*\alpha=\alpha$ basically means $\sum_{i=1}^n\xi_i dx_i=\sum_{i=1}^n\xi'_i dx'_i$. Writing $dx'_i$ in terms of $dx_i$ and $d\xi_i$, I've concluded that:
$$\sum_{i=1}^n\xi'_i\frac{\partial x'_i}{\partial x_j}=\xi_j$$
$$\sum_{i=1}^n\xi'_i\frac{\partial x'_i}{\partial \xi_j}=0$$
for every $j=1,…,n$. This seemed like a good start, but I'm not being able to conclude $dF\left(\sum_{i=1}^n\xi_i\frac{\partial}{\partial \xi_i}\right)=\sum_{i=1}^n\xi'_i\frac{\partial}{\partial \xi'_i}$. To arrive at that, I would need:
$$\sum_{i=1}^n\xi_i\frac{\partial x'_i}{\partial \xi_j}=0$$
$$\sum_{i=1}^n\xi_i\frac{\partial \xi'_i}{\partial \xi_j}=\xi_j$$
but I don't know how to get there. Any hints?
Best Answer
The canonical symplectic form $\omega$ identifies $\mathfrak{X}(T^*Q)$ and $\Omega^1 (T^*Q) $ by taking a vector field $Z$ to the $1$-form $Z_\flat =\omega(Z,\cdot)$. The idea is that the Euler field corresponds to the tautological form $\alpha$, perhaps modulo sign.
By the expression $\omega=\sum_{i=1}^n {\rm d}x^i\wedge {\rm d}\xi_i$, we get that $$\omega(X,\cdot) =\sum_{i=1}^n \begin{vmatrix} {\rm d}x^i(X) & {\rm d}x^i \\ {\rm d}\xi_i(X) & {\rm d}\xi_i\end{vmatrix}=\sum_{i=1}^n \begin{vmatrix} 0 & {\rm d}x^i \\ \xi_i& {\rm d}\xi_i\end{vmatrix} = -\sum_{i=1}^n \xi_i\,{\rm d}x^i =-\alpha.$$So $F^*\alpha = \alpha$ reads ${\rm d}F(X) = X\circ F$.
Further clarification: let $m\in T^*Q$ and $Y\in \mathfrak{X}(T^*Q)$. Then $F^*(-\alpha)_m (Y_m)=-\alpha_m (Y_m) $ becomes $$\begin{align} \omega_{F(m)}(X_{F(m)},{\rm d}F_m(Y_m)) &= -\alpha_{F(m)}({\rm d}F_m (Y_m)) \\ &= -\alpha_{m}(Y_m)\\ &=\omega_m (X_m,Y_m) \\ &= \omega_{F(m)}({\rm d}F_m (X_m), {\rm d}F_m(Y_m)).\end{align}$$Non-degeneracy of $\omega$ gives $X_{F(m)} = {\rm d}F_m (X_m)$.