show the Wronskian is constant

Question

Let $p,q : \Bbb{R} \to \Bbb{R^n}$ and $H:\Bbb{R^n}\times \Bbb{R^n} \to \Bbb{R} $

and the hamiltonian system:

$$ \begin{cases} \dot p = - \frac{\partial H}{\partial q} \\ \dot q = \frac{\partial H}{\partial p} \end{cases} $$

1. show that for p,q solutions of the system $ H(p,q)$ is constant over time
2. show that given that the system is linear then it's Wronskian is constant

I solved 1 by simply deriving $H(p,q)$ and getting that it's 0

for 2, I think that by using the formula $$ w(t) = w(t_0) e^{\int trA(s)ds} $$ I need to show that $tr(A )= 0$, but cant I manage to show it

Answer 1

Linear means that $H(p,q)$ is a quadratic form $H(p,q)= <Sp,p>+2<Tp,q>+ <Uq,q>$, $S,U$ symmetric.Then the matrix $A$ is the bloc matrix $ (-2T, -2U)$ on the first $n$ lines $(2S, 2T^t)$ on the last $n$ lines. Its trace is $0$, as $tr T=tr T^t$.