Linked Questions

2
votes
1answer
166 views

Interesting Hamiltonian System [duplicate]

The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function. I am wondering if ...
27
votes
3answers
2k views

Does topology have any role in classical physics?

I've seen many applications of topology in Quantum Mechanics (topological insulators, quantum Hall effects, TQFT, etc.) Does any of these phenomena have anything in common? Is there any intuitive ...
13
votes
3answers
1k views

What kind of manifold can be the phase space of a Hamiltonian system?

Of course it should have dimension $2n$. But any more conditions? For example, can a genus-2 surface be the phase space of a Hamiltonian system?
2
votes
4answers
951 views

Why is the phase space of a simple pendulum defined on a cylinder and not $\mathbb{T}^{2}$?

Let's take the pendulum equation $\ddot{x} = -\sin x$. Here $x \in \mathbb{T}^{1}$. Now rewrite it as a coupled first order system $$\dot{y} = -\sin x, \quad \dot{x}=y.$$ Intuitively we know that $y$ ...
6
votes
3answers
423 views

What are some mechanics examples with a globally non-generic symplecic structure?

In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be $\...
6
votes
3answers
261 views

Is there any Classical Mechanics system which needs to be described by a spinor?

We need an ordinary number (scalar) to describe a harmonic oscillator, and a vector to describe, for example, a pendulum. Is there any similarly simple system which we need to describe using a (two-...
4
votes
2answers
277 views

Phase space with torus topology

Consider a particular compact 2D symplectic manifold $\mathcal{M}$ defined as follows: The topology of $\mathcal{M}$ is a 2-torus. Let $\theta$ and $\varphi$ be the coordinate patch on $\mathcal{M}$ ...
2
votes
2answers
100 views

Is there a physical interpretation of symplectic manifolds which are not cotangent bundles?

The inspiration for symplectic geometry was from Hamiltonian mechanics. However, I am wondering how close the ties are between arbitrary symplectic manifolds and real physical systems. In ...
1
vote
1answer
134 views

Phase space as differential manifold

Generally we "draw" phase space as typical coordinate system, where $q$s and $p$s are treated like perpendicular axes. Why do we then regard phase space as generall differential manifold while it ...
0
votes
1answer
71 views

Physical meaning of theorem

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
0
votes
1answer
83 views

Do compact symplectic manifolds play a role in physics?

In classical mechanics, the phase space is the cotangent bundle of the configuration space, and it is a symplectic manifold, but not compact. Do compact symplectic manifolds have physical meaning? ...