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5
votes
1answer
547 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 ...
18
votes
5answers
6k views

What is the connection between Poisson brackets and commutators?

The Poisson bracket is defined as: $$\{f,g\}_{PB} ~:=~ \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial ...
9
votes
1answer
575 views

Understanding Poisson brackets

In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
7
votes
5answers
2k views

What does symplecticity imply?

Symplectic systems are a common object of studies in classical physics and nonlinearity sciences. At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context ...
3
votes
1answer
210 views

Geometric mechanics - Symplecticity

I am just trying to wade through literature on classical mechanics and I really don't know where to start, everything is Fibre bundle this or manifold that, and doesn't really ease you in to the ...
5
votes
2answers
458 views

Heisenberg picture of QM as a result of Hamilton formalism

Consider the formula for the total time-derivative of a physical value in Poisson's formalism: $$\tag{1} \frac{dA}{dt} = -\{H, A\}_{P.B.} + \frac{\partial A}{\partial t}, $$ where $\{A, B\}_{P.B.}$ is ...
7
votes
4answers
1k views

Connection between Poisson Brackets and Symplectic Form

Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ {q,p} $ basis are the same as those of the inverse of the symplectic matrix $ \Omega^{-1} $, whereas the matrix elements ...
6
votes
1answer
294 views

Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
4
votes
2answers
490 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 ...
3
votes
1answer
123 views

Yang-Mills constraints and Poisson brackets

Let's have constraints for Yang-Mills theory: $$ \varphi_{a} = \partial_{i}\pi^{i}_{a} - f_{abc}\pi^{b}_{i}A^{c}_{i}. $$ I have read the statement that $$ \tag 1 [\varphi_{a}(\mathbf x), ...
1
vote
2answers
506 views

Canonical equal time commutation relations in QED

I understand that to quantize the classical electromagnetic field one needs to impose commutation relations and express the field in terms of creation and annihilation operators. I notice that the ...
1
vote
2answers
433 views

Full time-derivative of a function and Schrodinger equation

From Hamiltonian formalism there is well known equation, $$ \frac{d F}{dt} = \frac{\partial F}{\partial t} + \{F, H\}_{PB}, $$ where $ \{H, F\}_{PB}$ is the Poisson bracket. After using Hamiltonian ...
1
vote
1answer
307 views

Why is $\{Q, P\} = 1$ for a canonical transformation?

Why is $\{Q, P\} = 1$ for a canonical transformation? Given $P(p,q)$ and $Q(p,q)$.
0
votes
1answer
82 views

Simple explanation of first and second class constraints with an example

Can someone give a simple physical example of first class and second class constraints? I mean, if you were giving a classical mechanics lecture for undergraduates, how would you explain this concept ...