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Injective flow lines of a vector field near a closed orbit

2011-03-07T09:23:17Z

Suppose $v$ is a vector field on a manifold $X$ with flow $\phi^t$. Suppose $v$ carries a first integral $f:X \rightarrow \mathbb{R}$ (i.e. $f$ is constant on the orbits of $v$). Suppose $\gamma:[0,T]\rightarrow X$ is a closed orbit of $v$, that is, $\gamma(x)=\phi^t(x)$ with $\phi^T(x)=x$. Suppose $f(\gamma)\equiv c$. Suppose $U \subset f^{-1}(c)$ is a compact neighborhood of $\mathrm{Im}(\gamma)$ in $f^{-1}(c)$ such that $U$ contains no other closed orbits of $v$ other than $\gamma$ (or it's translates).

What conditions are sufficient to impose on $v$ so that there exists some $t_* \ge 0$ such that no flow line of $v$ apart from $\gamma$ can live in $U$ for time longer than $t_*$. More precisely, if $\gamma_1$ is another flow line of $v$ such that $\mathrm{Im}(\gamma_1) \cap U \ne \emptyset$, and $[a,b] \subset \mathbb{R}$ is any interval such that $\gamma_1([a,b]) \subset U$ then $|b-a| \le t_*$.

Edit

(Thanks to Willie Wong)

Assume $X$ has dimension $\ge 3$. The question is perhaps better phrased as "are there well known conditions that always guarantee such a bound?".

Edit

(Thanks to Pietro Majer)

Sorry the question is badly phrased. I've tried to improve it. I'm most interested in the case where $v$ carries a first integral $f:X \rightarrow \mathbb{R}$ (i.e. $f$ is constant on the orbits of $v$), and we only care about orbits $\gamma_1$ such that $f(\gamma_1)=f(\gamma)$. I have thus rewritten the question. But the alternative question where the orbit $\gamma_1$ is required to have the same period as $\gamma$ is also interesting to me.

Contact manifolds that are not cooriented

2011-02-18T20:52:33Z

One always sees the definition of a contact manifold $(X,\xi)$ as an odd dimensional manifold with a hyperplane distribution $\xi$ which can locally be expressed as $\xi = \ker \alpha$ for a 1-form $\alpha$. But in fact, it seems that in every example I know of, one always assumes that $\xi$ is cooriented, and hence we can write $\xi = \ker \alpha$ globally.

Is there a reason (other than historical) as to why coorientation wasn't built in automatically in the definition of a contact manifold? It seems strange that this isn't required in the definition.

A $C^2$ small autonomous Hamiltonian has only constant 1-periodic orbits

2011-02-15T19:25:30Z

Consider a autonomous Hamiltonian $h:W\rightarrow \mathbb{R}$, where $W$ is a symplectic manifold. Let $\mathrm{sgrad}(h)$ denote the vector field on $W$ that is dual to the differential $Dh$ using the symplectic form.

Then it seems to be often stated that if $h$ is $C^2$ small then the only 1-periodic orbits of $\mathrm{sgrad}(h)$ are constants. Why is this?

Comment by

2011-02-15T19:34:19Z

thanks! that was very quick!