All Questions
Tagged with classical-mechanics sg.symplectic-geometry
29 questions
5
votes
1
answer
562
views
Definition of a moment map with physical context
This was originally posted on Math Stack Exchange, but without an answer. I thus move it here, and hope it's not because I express it unclearly.
Suppose $(M,\omega)$ is a symplectic manifold "well" ...
CommunityBot
- 1
1
vote
1
answer
125
views
isotropy of the cotangent lift of a group action
I asked this question in stack exchange but have not received an answer, so I am posting it here.
Given a group action on a manifold (e.g. configuration space of coordinates), cotangent-lift it to the ...
- 5,824
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
10
votes
0
answers
658
views
Determinant as a Hamiltonian
Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...
- 52.4k
20
votes
4
answers
3k
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 ...
CommunityBot
- 1
0
votes
1
answer
160
views
Reference for action-angle coordinates [closed]
Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !
- 63.9k
3
votes
1
answer
2k
views
Arnold's book on classical mechanics [duplicate]
Arnold's book “Mathematical methods of classical mechanics” develops the standard material on mechanics (e.g. the 3 Newton’s laws and the gravity law etc.). But what differs it from all other ...
- 91.8k
19
votes
3
answers
3k
views
Applications of symplectic geometry to classical mechanics
It is claimed that classical mechanics motivates introduction of symplectic manifolds. This is due to the theorem that the Hamiltonian flow preserves the symplectic form on the phase space.
I am ...
- 8,098
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
2
votes
0
answers
195
views
How to check conditions for Liouville-Arnold theorem? [closed]
Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem:
Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...
- 679
1
vote
1
answer
330
views
Global reduction of Hamiltonian with an integral of motion (Poincare' reduction)
This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question.
Background
I refer to the following concepts:
Liouville ...
- 1,301
7
votes
2
answers
2k
views
Practical example of Hamiltonian reduction
I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p_1, \...
- 31.5k
2
votes
0
answers
1k
views
Proof of Arnold-Liouville theorem in classical mechanics [closed]
I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point.
Especially, I am talking about the part that ...
- 5,293
24
votes
1
answer
2k
views
Why are Lagrangian submanifolds called Lagrangian?
Much of the terminology in symplectic geometry comes from classical mechanics: the symplectic manifold is modeled on a cotangent bundle $T^*N$ of some configuration space $N$ with local position ...
- 31.5k
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 ...
- 366
2
votes
0
answers
285
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 ...
- 4,306
7
votes
0
answers
479
views
Question about theorem in Arnold's book on action-angles variables
I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question)
If you don't have the book or ...
- 259
3
votes
1
answer
355
views
Local symplectomorphisms become global ones?
It is widely known that a local diffeomorphism is not necessarily a global diffeomosphism and so on.
Now, I stumbled over the question whether in some particular cases, as I will describe below, ...
- 3,377
0
votes
1
answer
304
views
Lagrangian flow preserves symplectic form
Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...
- 685
7
votes
2
answers
2k
views
Momentum a cotangent vector
Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.
Furthermore, many sources ...
- 231
2
votes
0
answers
165
views
Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom
Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation
$$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$
can ...
- 1,038
4
votes
1
answer
396
views
Weinstein's local classification of Lagrangian foliations
In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I ...
- 576
6
votes
3
answers
450
views
Do there exist small neighborhoods in a classical mechanical system without pairs of focal points?
The question I will ask makes sense in much more generality, but I will leave the translation to the experts, since I'm only looking for a special case (and it would not surprise me if the answer does ...
- 19.4k
3
votes
2
answers
589
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.
...
- 1,710
1
vote
1
answer
1k
views
Generating functions and Lagrangian submanifolds
I'm interested in showing the existence of a generating function. Explicitly:
Suppose $\Lambda\subset T^*M\times T^*M$ is a Lagrangian submanifold. Consider the projection $\pi:(x_1,\xi_1,x_2,\xi_2)...
- 4,306
9
votes
1
answer
596
views
Classical analogue of the Stone-von Neumann Theorem?
Let $U_s$, $V_t$ be a pair of continuous $n$-parameter groups ($n < \infty$) of unitary operators on a complex Hilbert space $\mathcal{H}$. The Stone-von Neumann Theorem establishes that any such ...
- 8,336
2
votes
0
answers
356
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 $...
- 614
8
votes
1
answer
2k
views
Calculating the geodesic equation for a particular set of phase-space coordinates
Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ ...
- 108k
24
votes
3
answers
3k
views
Classical mechanics motivation for poisson manifolds?
Suppose I want to understand classical mechanics.
Why should I be interested in arbitrary poisson manifolds and not just in symplectic ones?
What are examples of systems best described by non ...
- 4,577