Again banged my head trying to make sense of the Mechanics course and hit a rock bottom with another problem. I would really appreciate the step by step explanation for the results of it.
We have a symplectic manifold (M, ω), a Hamilton function H : M -> $\mathbb{R}$ and c a regular value of H and consider the level $E_c := H^{-1}(c)$.
a) Show that $E_c$ is a manifold and give its dimension. Also show that for any $x\in E_c$ the tangent space at $E_c$ is given by $T_xE_c = kerdH_x$
From this we are given the following facts: let the Liouville volume 2n-form
$\Omega$:= $\omega \wedge \omega \wedge...\wedge \omega$ the n-fold
wedge product. The flow $\phi^t$ of the Hamiltonian vector field $X_H$ preserves
$\Omega$. Also, the level $E_c$ is preserved by the same $\phi^t$.
,Consider $x\in E_c$ and tangent vectors $\xi_1, \xi_2; ...\xi_{2n-1} \in T_xE_c$; having to define $ \Omega_{c,x}(\xi_1, \xi_2; ...\xi_{2n-1})$
To this end we write the equation
$ \Omega_{x}(\eta, \xi_1, \xi_2; ...\xi_{2n-1}) = dH_x(\eta)\wedge \Omega_{c,x}(\xi_1, \xi_2; ...\xi_{2n-1}) $, $\eta \in T_xM$ is arbitrary.
b)Show that this equation determines $\Omega$ in a unique way, independent of $\eta$
c)Also show that $\Omega_c$ is a nondegenerate (2n-1) form, i.e. a volume form on $E_c$
d) Finally show that $\phi^t|E_c$ preserves $\Omega_c$