All Questions
Tagged with hamiltonian-mechanics sg.symplectic-geometry
22 questions with no upvoted or accepted answers
11
votes
0
answers
233
views
Mathematical pendulum and $\mathbb C P^n$
I am very puzzled by the following remark on p.346 in Arnold's book "Mathematical methods of classical mechanics":
Another method of construction the same symplectic structure on complex ...
- 5,063
8
votes
1
answer
351
views
From time-dependent Hamiltonians to time-dependent symplectic/Poisson structures
Let $(M,\{.,.\})$ be a smooth Poisson manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.
Question: Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and smooth parameter-dependent Poisson ...
- 1,959
7
votes
0
answers
144
views
Reference request: Liouville integrability of a torus action of small dimension on a symplectic manifold
Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by $k$, may be less than $n$. The generators of the action will ...
- 4,787
6
votes
0
answers
537
views
Hamiltonian dynamics on cotangent bundle
I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
- 341
6
votes
0
answers
469
views
An algebraic Hamiltonian vector field with a finite number of periodic orbits (2)
Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
- 366
5
votes
0
answers
274
views
Deformation quantization of Poisson bracket without star-product
Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...
- 654
3
votes
0
answers
137
views
When the integrable system defines a moment map?
Assume $\mathcal{M}$ is a compact symplectic $2n$-dimensional manifold with a Hamiltonian action of the torus $\mathbb{T}^n$. Given a family of functions $F=(f_1,\ldots,f_n)$ defining an integrable ...
3
votes
0
answers
143
views
Is composition of discrete Hamiltonian flows integrable?
Consider $\Bbb{R}^2$ with the usual symplectic form $$\omega = dx \wedge dy$$
For a function $H \colon \Bbb{R}^2 \to \Bbb{R}$, let $X_H$ be the Hamiltonian vector field. Then the map $\Bbb{R}^2 \to \...
- 213
3
votes
0
answers
68
views
Infinitesimal orbit type decomposition of Hamiltonian $G$-manifolds
Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. ...
- 1,942
2
votes
0
answers
153
views
Spectrum of an almost Hamiltonian matrix
I have a complex-valued block matrix $N=\begin{bmatrix}
A & B \\
C & -A^*
\end{bmatrix}$, where $A$ is diagonal, $B=B^*$, and $C$ is rank-1 but not Hermitian.
If $C$ were Hermitian, $N$ would ...
- 116
2
votes
0
answers
53
views
Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials
I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses:
Coulomb potential with a ...
- 21
2
votes
0
answers
129
views
Is there a notion of symplectic maps between spaces of volume forms on phase spaces?
For a $n$ dimensional smooth manifold $M$, I consider the cotangent bundle $T^*M$ with the canonical symplectic form $\omega$.
A symplectic map $\phi : T^*M \to T^* M$ is a map which leaves the ...
2
votes
0
answers
192
views
What is the relation between the different generating functions thought as finite approximations of action functionals
In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on $\mathbb{R}^{2n}$. The idea is that a Hamiltonian isotopy can be ...
- 1,227
2
votes
0
answers
100
views
Effective actions by non-commutative groups have non-commuting fundamental vector fields?
I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :)
Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ ...
- 989
2
votes
0
answers
491
views
How to make sense of the Euler Lagrange equations for an infinite action?
The Euler–Lagrange equation is an equation satisfied by a function $q$, which is a stationary point of the functional
$S(\boldsymbol q) = \int_a^b L(t,q(t),\dot{q}(t))\, \mathrm{d}t$
Say we have an ...
- 989
2
votes
0
answers
129
views
Is the interpolating Hamiltonian flow of an exact near-identity symplectic map globally defined?
It is well-known that an analytic near-identity map $\bar{x} = F_{\epsilon}(x) = x + \epsilon f(x) + O(\epsilon^{2})$ may be embedded into the flow of a differential equation, and if that map is ...
- 21
2
votes
0
answers
143
views
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 ...
- 197
1
vote
0
answers
77
views
Projecting phase space Liouville tori to configuration space in integrable systems
My question concerns whether or not the topology of a trajectory in the configuration space of an integrable system, or rather the area of configuration space accessible by trajectories can be ...
- 11
1
vote
0
answers
44
views
When lagrangian fibrations are equivalent?
Given a $2n$-dimensional symplectic manifold $\mathcal{M}$ and two different lagrangian fibrations $\pi_1:\mathcal{M}\rightarrow \Gamma_1$ and $\pi_2:\mathcal{M}\rightarrow \Gamma_2$, with $\Gamma_1, \...
1
vote
0
answers
79
views
Liouville-Arnold and fibration relative to a convex polytope
Liouville-Arnold's theorem indicates that given a Hamiltonian torus action on a manifold and a set of $n$ functions $F$ from the manifold to $\mathbb{R}^n$ defining an integrable system, the pre image ...
0
votes
0
answers
110
views
Substitution of the old variable to a generating function
Assume that we have one of the 4 types of generating functions (say F(p, Q)) which defines a symplectomorphism $T$:
$(P,Q) = (u(p,q), v(p,q))$. Now we can make a substitution and get a function $G(p,q)...
- 728
0
votes
0
answers
94
views
On the measure of regular and chaotic regions in a phase space
Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...
- 255