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16
votes
5answers
5k 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 ...
5
votes
1answer
400 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 ...
7
votes
5answers
1k 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 ...
5
votes
2answers
380 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 ...
7
votes
4answers
834 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 ...
3
votes
1answer
153 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 ...
4
votes
2answers
450 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
80 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
395 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
364 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
263 views

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

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