2
votes
1answer
142 views

Question about Lie derivative

$(M,w)$ is symplectic manifold. $f_t : M\to M$ is a symplectic isotopy between $f_0=id$ and $f_1$. Let X_t be the vector field on M satisfying $d(f_t)/dt=X_t(f_t)$ Now I differentiate $(f_t)^*w$. Here ...
1
vote
1answer
53 views

Can the system $\partial_x f(x,y) = \dot{y}$, $\partial_y f(x,y) = \dot{x}$ be related to some Hamiltonian system?

If one has found some function $f(x,y): \partial_x f = \dot{y}, \partial_y f = \dot{x}$, is there a simple transformation or change of variables that results in Hamilton's equations $\partial_p H = ...
1
vote
2answers
245 views

Application of the implicit function theorem

Assume that the equation $F(x,y,p)=0$ defines a regular submanifold $M$ of $R^3$. Consider the projection $\pi :M \rightarrow R^2$, given by $\pi (x,y,p)=(x,y)$. By the implicit function theorem, in ...
2
votes
2answers
133 views

Non-degenerate solutions to constant Hamiltonian flow

As I'm trying to work my way through Dietmar Salamon's "Notes on Floer Homology", I'm having trouble with the very first exercise. Let $(M, \omega)$ be a compact symplectic manifold. Let $H$ be a ...
3
votes
4answers
108 views

reference for strongly continuous semi-groups

At the moment I am trying to understand the proof of the Fredholm property in Salamon's notes on Floer homology. There I came across the notion of an unbounded operator on a (real) Hilbert space which ...
1
vote
1answer
310 views

Harmonic Oscillator and Quadrature

Consider the simple harmonic oscillator $\frac{d^2p}{dt^2}=-p$ as a Hamiltonian system with Hamiltonian given by $H=\frac{1}{2}p^2+\frac{1}{2}q^2$. The famous Liouville theorem for integrable systems ...