4
$\begingroup$

Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property:

For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X_H$ on $H^{-1}(a)$ is not topological equivalent to the dynamic of $X_H$ on $H^{-1}(b)$?

Here $X_H$ is the hamiltonian vector field associated to the Hamiltonian $H$.

The motivation for this question is the following post. The answer to the following post would be negative if the answer to the above question is negative:

An algebraic Hamiltonian vector field with a finite number of periodic orbits (2)

Improve this question
$\endgroup$
22
  • 1
    $\begingroup$ Do you assume the Palais-Smale condition on the function $H$? (e.g., $H$ is coercive). Otherwise in your hypotheses already the sub-manifolds $H^{-1}(a)$ and $H^{-1}(b)$ may well differ topologically (for instance: one is empty, the other is not, $H$ is bounded below but does not have a minimum) $\endgroup$
    – Pietro Majer
    Commented Jul 23, 2023 at 16:53
  • 2
    $\begingroup$ There is a "critical point at infinity" at the level c=0, that is a sequence with $f(x_n)\to c$, $df(x_n)\to0$, $x_n\to\infty$ (in the 1 pt compactification). It can be an obstruction to deformation just as a true critical point $\endgroup$
    – Pietro Majer
    Commented Jul 24, 2023 at 7:03
  • 2
    $\begingroup$ For another example: take any smooth $f:M\to\mathbb R$ with two non empty non diffeomorphic regular level sets with values a and b. Then remove the critical set $\text{crit}(f):=\{x\in M: df(x)=0\}$ from $M$: now $[a,b]$ consists of regular values of $g:=f$ restricted to $N:=M\setminus\text{crit}(f)$, since $g$ has no critical points. (And of course, if $\text{crit}(f)$ was non-empty, $g$ does not satisfy PS). $\endgroup$
    – Pietro Majer
    Commented Jul 25, 2023 at 17:27
  • 2
    $\begingroup$ I did not recalculate this, but I think the following is true: even if you do not have compactness issues the dynamics can change drastically. You can have a hamiltonian on $R^4$ whose regular level set at $1$ is a sphere, where all orbits are closed, but at $2$ it is an ellipse with very few closed orbits. $\endgroup$
    – Thomas Rot
    Commented Jul 27, 2023 at 12:03
  • 2
    $\begingroup$ Here the sphere is $\mathbb S^3$, all solutions are great circles. His Hamiltonian could even be a strictly convex coercive function I think. Very nice $\endgroup$
    – Pietro Majer
    Commented Jul 27, 2023 at 20:01

1 Answer 1

Reset to default
4
$\begingroup$

This is again with the caveat that I have not done the calculations. Let $a>1$ be an irrational number. Let $H:\mathbb{R}^4\rightarrow \mathbb R$ be a function such that

$ H(x,y,z,w)=(x^2+y^2+z^2+w^2) $

when $x^2+y^2+z^2+w^2=1$ and

$ H(x,y,z,w)=x^2+y^2+z^2+aw^2 $

when $x^2+y^2+z^2+aw^2=2$. You can convince yourself that it is possible to find such a function without critical points in the region

$$ 1\leq x^2+y^2+z^2+w^2\qquad \text{and}\qquad x^2+y^2+z^2+aw^2\leq 2 $$

(i.e. by interpolating the values in the region in between radially)

At the level set $H^{-1}(1)$ all orbits are closed because these are two uncoupled harmonic oscillators with the same period. At the level $H^{-1}(2)$ the motion is also described by two harmonic oscillators, but now with periods that are irrational with respect to each other. The only periodic orbits are those where one of the oscillators is not in motion. So there are two of them.

Improve this answer
$\endgroup$
3
  • $\begingroup$ My +1 and thanks for your attention to my question and your answer. I think some thing is missing. It seems that you actually use the following statement but I think it is not true: "If $N$ is codimension one submanifold and $H_1$ and $H_2$ are constant on $N$ then the two Hamiltonian flow are the same on $N$" . This is not true for example put $N=$ unit circle in the plane, $H_1\equiv 0$ and $H_2=x^2+y^2-1$ The first Hamiltonian consist of singularities and the second one consist of a periodic orbit, two different dynamic. Am I missing some thing? $\endgroup$
    – Ali Taghavi
    Commented Jul 28, 2023 at 10:34
  • 1
    $\begingroup$ I use the following fact: If $N$ is a regular level set of two different Hamiltonians $H_1$ and $H_2$, then the dynamics on $N_1$ with respect to $H_1$ is a rescaling of the dynamics of $H_2$ on $N$. $\endgroup$
    – Thomas Rot
    Commented Jul 28, 2023 at 10:59
  • 1
    $\begingroup$ See for example the discussion in Weinstein's On the hypotheses of Rabinowitz' periodic orbit theorems $\endgroup$
    – Thomas Rot
    Commented Jul 28, 2023 at 11:02

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .