The poisson-brackets tag has no wiki summary.
7
votes
4answers
2k views
What is the connection between Poisson brackets and commutators?
The Poisson bracket is defined as:
$$\{f,g\} = \sum_{i=1}^{N} \left[
\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} -
\frac{\partial f}{\partial p_{i}} \frac{\partial ...
6
votes
5answers
545 views
What does symplecticity imply?
Symplectic systems are a common object of studies in classical physics and nonlinearity sciences.
At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context ...
5
votes
3answers
237 views
Poisson structure comes from hamiltonian?
I am interested in studying quantization, but it seems I am lacking the basics of classical mechanics. Any help would be appreciated.
I would first like to ask what is necessary to have a ...
4
votes
2answers
194 views
The string Poisson bracket
Where does the factor $\frac{1}{T}$ ($T$ is the string tension) in this Poisson bracket come from?
$$ \{X^{\mu}(\tau,\sigma),\dot{X}^{\nu}(\tau,\sigma')\} ~=~ ...
4
votes
2answers
108 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 ...
4
votes
1answer
162 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 ...
3
votes
3answers
116 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 ...
2
votes
2answers
449 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 ...
2
votes
1answer
158 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 ...
1
vote
1answer
324 views
Poisson brackets and angular momentum
I'm trying to find $[M_i, M_j]$ Poisson brackets.
$$\{M_i, M_j\}=\sum_l \left(\frac{\partial M_i}{\partial q_l}\frac{\partial M_j}{\partial p_l}-\frac{\partial M_i}{\partial p_l}\frac{\partial ...
1
vote
1answer
116 views
Canonical transformation and Hamilton's equations
I was trying to prove, that for a transformation to be Canonical, one must have a relationship:
$$
\left\{ Q_a,P_i \right\} = \delta_{ai}
$$
Where $Q_a = Q_a(p_i,q_i)$ and $P_a = P_a(p_i,q_i)$.
Now ...
