2
votes
0answers
21 views

Some question about symplectic transformation

I read Arnold's book Mathematical Methods of Classical Mechanics and come across with three problems in page 229. 1.Let $\lambda$ and $\bar{\lambda}$ be simple (multiplicity 1) eigenvalues of a ...
3
votes
1answer
80 views

A question about canonical transformation

I have posted this question in math.stackexchange before with no answer till now. It may be more suitable to post here. There is a problem in Arnold's Mathematical Methods of Classical Mechanics ...
4
votes
1answer
43 views

Why is the Legendre transformation an application of the duality relationship between points and lines?

When I read the Wiki about Legendre transformation, there is a statement The Legendre transformation is an application of the duality relationship between points and lines. What's the meaning of ...
0
votes
1answer
26 views

A question about Hamiltonian phase flow

Show that if a one-parameter group of difeomorphisms of a symplectic manifold preserves the symplectic structure then it is a locally hamiltonian phase flow. Note that A locally hamiltonian ...
1
vote
1answer
89 views

What makes an abstract physical system describable by a “fluid” equations of motion?

We can describe (some of) the dynamics of many systems using fluid mechanics. Of course these include classical fluids like water, more exotic fluids like photon gases and the universe as a whole and ...
0
votes
0answers
66 views

Liouville's theorem on integrable Hamiltonian systems is a let down

I read a proof of Liouville’s theorem on integrable Hamiltonian systems. According to the theorem, an autonomous Hamiltonian system can be integrated in quadratures, given $n$ involutive first ...
2
votes
3answers
213 views

Probability density in Hamiltonian Mechanics

I am currently studying Liouville's theorem compare wikipedia and there this mysterious probability density $\rho$ appears and I was wondering how one can determine this quantity analytically for a ...
1
vote
0answers
108 views

How to understand the matrix behind a Hamiltonian?

thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be ...
2
votes
1answer
124 views

Does a constant of motion always imply a Hamiltonian formulation?

If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
2
votes
4answers
498 views

Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
2
votes
1answer
191 views

A good example of a nonlinear symplectomorphism?

What is a good example of a simple, physically useful nonlinear symplectomorphism $\kappa: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$? I'm not much of a physicist, and all the examples I've worked ...
6
votes
2answers
432 views

How do we resolve operator ordering ambiguities when quantizing generic nonlinear second-class constraints?

Dirac came up with a general theory of constraints, including second-class constraints. To quantize such systems, he first computed the Dirac bracket classically, and only then "promoted" the ...