This question has come out while reading J. Moser "*New Aspects in the Theory of Stability
of Hamiltonian Systems*". I'm particularly interested to the Appendix, where one investigates the stability of elliptic fixed points of Hamiltonian dynamical systems, in the time independent case. I start presenting the **framework**.

Let us consider the Hamiltonian dynamical system $$ \dot{x}_\nu = H_{y_\nu}(x,y), \qquad \dot{y}_\nu=-H_{x_\nu}(x,y) \qquad \qquad (1) $$

where $\nu=1,2,\dots,n$. Hamiltonian $H$ does not depend on time $t$ and it is assumed to be a real analytic function of $x_\nu,\,y_\nu$, with $\nu=1,2,\dots,n$ in the neighboorhood of $x=y=0$, the expansion of which starts with quadratic terms. Then $x=y=0$ is an equilibrium solution.

One can construct a fundamental system of solutions of exponential form $$ w^{(\nu)}=e^{\gamma_\nu t}p^{(\nu)} $$ where $p^{(\nu)}$ are constant vectors or, in the case of multiple eigenvalues $\gamma_{\nu}$, possibly polinomials in $t$. The numbers $\gamma_\nu$ are obtained as the eigenvalues of the matrix determined by the linear terms of the right-hand side of (1). Suppose that all eigenvalues are distinct and purely imaginary, i.e. of the type $\gamma_\nu=i\beta_\nu$, with $\beta_\nu$ real. So the spectrum has the form $$ \pm i \beta_1, \quad \pm i \beta_2, \,\dots,\, \pm i \beta_n. $$ So one obtains a collection of distinct numbers $\beta_\nu$, $-\beta_\nu$ with $\nu=1,2,\dots,n$

So far, so good.

For later purposes, one needs to define the sign of $\beta_\nu$. The Author says that the sign of $\beta_\nu$ is taken in such a way that $$ \mathcal{Im}\left[w^{(\nu)},\overline{w^{(\nu)}}\right]<0. $$ Square brackets are defined as Lagrange Brackets (an outdated therminology, nowadays called symplectic form [see comments]). More precisely, given any two $2n$-dimensional vectors, $x$ and $\tilde{x}$, with components $x_\nu$ and $\tilde{x}_\nu$, their Lagrange Bracket (read: symplectic form) is defined as $$ [x,\tilde{x}]=\sum_{\nu=1}^n (x_\nu\tilde{x}_{\nu+n}-x_{\nu+n}\tilde{x}_\nu) $$ In passing, one has to remember that an Hamiltonian sysytem is marked by the fact that, for any two solutions $x$ and $\tilde{x}$, the Lagrange bracket (read: symplectic form) $[x,\tilde{x}]$ is $t$-independent.

**Questions:** can someone please show

1) How to explicitly compute:

$$ [w^{(\nu)},w^{(\mu)}] = ? $$

$$ [w^{(\nu)},\overline{w^{(\mu)}}] = ? $$

2) How to prove that $$ [w^{(\nu)},\overline{w^{(\nu)}}] = [p^{(\nu)},\overline{p^{(\nu)}}] $$

$$ [p^{(\nu)},\overline{p^{(\nu)}}] \text{ is purely imaginary} $$ 3) How to define the sign $\beta_{\nu}$ in such a way that $$ \mathcal{Im}\left[w^{(\nu)},\overline{w^{(\nu)}}\right]<0. $$ I am from the Physics community, so I kindly ask to display all important passages.