# Tagged Questions

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### Is it true that a nondegenerate minimizing periodic orbit of mechanical Hamiltonian system is hyperbolic

Consider mechanical Hamiltonian system of the form
$$H(p,q)=\dfrac{\Vert p\Vert^2}{2}+V(q),\quad (q,p)\in T^*\mathbb T^n.$$
Here we suppose the periodic orbit $\gamma$ minimizes the Lagrangian ...

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**1**answer

266 views

### An algebraic Hamiltonian vector field with a finite number of periodic orbits

Edit: There is an interesting complete answer for the second part(see the answer by Thomas Kragh). I search for an answer for the first part.
1.Is there a polynomial Hamiltonian ...

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342 views

### Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...

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**1**answer

558 views

### What is known about the strong Arnold conjecture?

Here are the two versions of Arnold's conjecture on Hamiltonian orbits:
Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a nondegenerate ...

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**3**answers

477 views

### Given a vector field all of whose integral curves are closed, is the period a smooth function?

Disclaimer: The original question consisted of two parts. The first one
has been answered negatively (see
below the answers of Sam Lisi and
Alejandro). It remains the second one.
Background
...

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245 views

### Transversality and isolated degenerate critical points

Maybe some of the following statements are not precise. Please correct them.
Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ ...

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**2**answers

385 views

### How to deal with the singular reduction of the Hamiltonian n body problem?

I would like to consider the reduced Hamiltonian $n$ body problem, but am struggling with the angular momentum reduction seeing as the $SO(3)$ action is not free and the reduction is singular.
...

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285 views

### Symplectic submanifolds and first integrals

I was working with symplectic submanifolds when I posed the following question:
Suppose I have a Hamiltonian system with the phase space $\mathcal{M}$, a symplectic manifold with the standard ...

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1k views

### Flow of a Hamiltonian vector field

Smooth vector fields are in a one-to-one relationship with flows $\Phi: D \subseteq M\times \mathbb{R} \rightarrow M$,
$$X_m = {\frac{d}{d t}}_{t=0} \Phi(m, t),$$
and by the symplectic form also with ...

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**1**answer

377 views

### is the geodesic flow on Hyperbolic Plane completely integrable?

I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and ...

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294 views

### flexibility of almost contact ``Reeb'' vector fields

New version of the question:
Given an odd dimensional manifold $V$, an almost contact structure is a pair of $(\alpha, \omega)$, where $\alpha$ is a non-vanishing 1-form and $\omega$ is a 2-form ...

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**1**answer

417 views

### Generalization of Rigid Body Motion to arbitrary (compact) Lie Groups

The classical dynamics of a rigid body in three dimensions may be described as the motion of a point on a configuration space given by the Lie group $SO(3)$, governed by Euler's equations for rigid ...

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247 views

### Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...

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**1**answer

2k views

### Quantum dynamics on varieties and Salmon Prizes

Concluding Progressive Remarks
A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize.
The Salmon Prize (photo of the ...

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210 views

### In search for a more geometric proof of a result of van der Schaft and Maschke on nonholonomic mechanics.

Edit: Now I have found something that appears to answer my own question. It is section 2 in the paper "On Submanifolds and Quotients of Poisson and Jacobi Manifolds" by Ch.-M. Marle. (There, he ...

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237 views

### Dissipative Hamiltonian System with a Periodic Force

Let $H:P \to \mathbb{R}$ be a Hamiltonian on a symplectic manifold $(\omega,P)$ and let $X_H: P \to TP$ be the Hamiltonian vector-field. Let $F:P \to T^*P$ be a dissipative force field such that for ...

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**1**answer

546 views

### How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?

What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...

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**4**answers

1k views

### What is the role of contact geometry in the hamiltonian mechanics?

Let us assume someone is interested in the study of Hamiltonian mechanics.
What are good examples to illustrate him of the usefulness of contact geometry in this context?
On one hand the Hamiltonian ...

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**1**answer

429 views

### On degenerate integrable hamiltonian systems

Is there some reference where the existence of local generalized action-angle variables is discussed in some detail for concrete examples of hamiltonian systems of mechanical type?
After Dazord and ...

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**2**answers

417 views

### The fibers of the momentum map for the $SO(n+1)$ symmetry of the geodesic flow on $S^n$

My question is: Are the orbits of the geodesic flow on $S^n$ determined as the fibers of the momentum map for its $SO(n+1)$ symmetry?
I started by considering the analog problem for the orbits of the ...

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238 views

### Is an immersed Kronecker join always a multilinear variety on a Hilbert space?

The question asked is:
Is the implicitization of an arbitrary-rank immersed Kronecker join always a multilinear variety on a Hilbert space?
This is related to another MathOverflow question
...

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**2**answers

749 views

### On the proof of the hamiltonian flow box theorem

The hamiltonian flow box theorem, as stated in Abraham and Marsden's Foundations of Mechanics, says that:
Given an hamiltonian system $(M,\omega,h)$ with $dh(x_0)\neq 0$ for some $x_0$ in $M$, there ...

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793 views

### When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold?

Here is my precise question. Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function. The symplectic form gives rise to an isomorphism between the tangent ...

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### Floer homology and status of the Arnold conjecture.

The Arnold conjecture on a closed symplectic manifold $(M,\omega)$ says in the weakest version that for a non-degenerate Hamiltonian there are at least $k$ 1-periodic orbits where $k$ is the sum of ...

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**2**answers

827 views

### Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$.
I found this in claim a ...

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798 views

### What are some conserved quantities of Poisson brackets?

Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets.
Let's say we are working on T^n x R^n (T^n ...