3
votes
1answer
74 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 ...
0
votes
1answer
161 views

Calculate integral of motion condition with Poisson brackets

Problem statement: The Hamiltonian of a system is given by the formula: \begin{equation*} H = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2} + V(r,\theta). \end{equation*} Under what condition is ...
4
votes
2answers
313 views

Heisenberg picture of QM as a result of Hamilton formalism

Let's have formula of full time-derivative of physical value in Poisson's formalism: $$\tag{1} \frac{df}{dt} = -[H, f]_{P. br.} + \frac{\partial f}{\partial t}, $$ where $[A, B]_{P. br.}$ is Poisson's ...
1
vote
2answers
912 views

How to find out whether a transformation is a canonical transformation?

We had a couple of examples where we were supposed to calculate the Canonical Transformation (CT), but we never actually talked about a condition that decides whether a transformation is a canonical ...
3
votes
1answer
155 views

Two components of angular momentum conserved $\Rightarrow $ All three components are conserved?

I was wondering whether it is correct to say that if two components of the angular momentum are conserved, then all three Cartesian coordinates of the angular momentum are conserved? I would regard ...
2
votes
0answers
165 views

The correspondence between Poisson bracket and Commutators in Quantum Mechanics

I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this. How can we generally show that ...
5
votes
3answers
774 views

Physical interpretation of Poisson bracket properties

In classical Hamiltonian mechanics evolution of any observable (scalar function on a manifold in hand) is given as $$\frac{dA}{dt} = \{A,H\}+\frac{\partial A}{\partial t}$$ So Poisson bracket is a ...
6
votes
2answers
187 views

Are Poisson brackets of second-class constraints independent of the canonical coordinates?

Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\{\chi_N(q,p),\chi_M(q,p)\}_P$ . Is ...
3
votes
1answer
265 views

Find the Hamiltonian given $\dot p$ and $\dot q$

I have these equations: $$\dot p=ap+bq,$$ $$\dot q=cp+dq,$$ and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$. To answer to the first ...
4
votes
1answer
253 views

Clarifications about Poisson brackets and Levi-Civita symbol

I need some clarifications about Poisson brackets. I know canonical brackets and the properties of Poisson Brackets and I also know something about Levi-Civita symbol (definition and basic ...
1
vote
1answer
844 views

Poisson brackets and angular momentum

I'm trying to find $[M_i, M_j]$ Poisson brackets. $$\{M_i, M_j\}=\sum_l \left(\frac{\partial M_i}{\partial q_l}\frac{\partial M_j}{\partial p_l}-\frac{\partial M_i}{\partial p_l}\frac{\partial ...
2
votes
2answers
1k views

Poisson brackets: prove that they are canonical invariants

EDIT: I haven't forgotten to accept answer, the question is still open.. I need a clarification about Poisson brackets. I'm studying on Goldstein's Classical Mechanics (1 ed.). Goldstein proves ...
4
votes
2answers
327 views

Classical Limit of Commutator

In Dirac's book Principles of quantum mechanics (4th ed., pgs 87-88), he seems to give a very elementary argument as to how the commutator $[X,P]$ reduces to the Poisson brackets ${x,p}$ in the limit ...
6
votes
3answers
307 views

Poisson structure comes from hamiltonian?

I am interested in studying quantization, but it seems I am lacking the basics of classical mechanics. Any help would be appreciated. I would first like to ask what is necessary to have a ...