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Questions tagged [poisson-brackets]

In phase-space classical Hamiltonian mechanics, the Poisson bracket is an antisymmetric binary operation acting like a derivative.

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Transformation between a dynamical system to a Hamiltonian system [duplicate]

Consider a dynamical system characterized by these equations $$\dot{x}=x-xy \\ \dot{y}=-y+xy$$ If we transform $\ln(y)=q$ and $\ln(x)=p$, the system can be changed into a Hamiltonian system with $q$ ...
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Verifying completeness of constants of motion

I can find constants of motion by looking at the null space of the Poisson Bracket operator $\{H, \cdot\}$ over a polynomial space by brute force with symbolic algebra (code). This scales terribly ...
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If Poisson Bracket of Momentum and Position is non-zero, why no Uncertainty Principle?

In Hamiltonian classical mechanics, we have that the Poisson bracket of position and momentum satisfies $$\{q_i, p_j\} = \delta_{ij}$$ But this implies that momentum and position 'generate' changes ...
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Why do we care only about canonical transformations?

In Hamiltonian mechanics we search change of coordinates that leaves the Hamilton equation invariant: these are the canonical transformations. My question is: why we want to leave the equations ...
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Some basics about Bracket Notation

I'm trying to prove something. Sorry this post is so long but I wanted to keep things as basic as possible so people have an easier time understanding. Let's assume we have a quantum system $\rho$ ...
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Quantizing first class constraints

Let $\gamma$ denote a first class constraint. Then if there exists a function on phase space $f(q,p)$ for which the Poisson bracket with the constraint does not vanish $\lbrace f, \gamma\rbrace \neq 0$...
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Does the Heisenberg equation for fields and canonical momentums hold as well for the hamiltonian density operator instead of the Hamiltonian operator?

In quantum field theory, with the field $\phi$ and the momentum $\pi$ being operators, their time evolution is governed (in the Heisenberg-picture) by the Heisenberg equation: \begin{align} \dot{\phi}...
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Hamiltonian formalism and the phase space

In my book, it says that Hamilton's equations of motion are equations of the first order in the time and that they describe the motion of the system in the $2S$-dimensional phase space. Could someone ...
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Use of generating function in canonical transformation

In the theory of Canonical transformations, initially we use the fact that the new and the old system of $(q_i, p_i)$ with the Hamiltonian $H$ satisfy the modified Hamilton's principle. Now here, the ...
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Phase space volume doesn't change under canonical transform

I have given a set of generalized coordinates $(q_1,..q_n,p_1,..p_n)$. Suppose I had a canonical transform $(q_i,p_i)\rightarrow (Q_i,P_i).$ I am trying to show that the phase space volume element ...
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Poisson Bracket for the angular momentum and Runge-Lenz vector [closed]

Given the Runge-Lenz vector $$\vec{A}=\vec{p}\times\vec{L}-mk\frac{\vec{r}}{r}$$ and the angular momentum $$\vec{L}=\vec{r}\times\vec{p}$$ We can get $$\{L^i,A^j\}=\epsilon^{ijk}A^k.$$ I think ...
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Symplectic reduction to moduli space in Chern-Simons theory

In QFT and the Jones polynomial, Witten claims that it is possible to perform symplectic reduction from the distributional Poisson bracket on the unconstrained phase space to a symplectic structure on ...
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Identify a Hamiltonian system consistent or not?

I'm sorry if my question is too classic and basic. As Dirac-Bergmann algorithm for Hamiltonian formalism, I find out that a Hamiltonian system is inconsistent if Poisson bracket of primary constraints ...
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Hamiltonian under Canonical transformation

We know that 'Hamilton's equations' preserve in canonical transformation. But does this mean than 'Hamiltonian' itself doesn't change under canonical transformation?
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Star Product and Poisson Brackets

I have the following definition of star product, \begin{equation} \star=\exp\left[\frac{i\hbar}{2}\left(\frac{\overleftarrow{\partial}}{\partial Q^{I}}\frac{\overrightarrow{\partial}}{\partial P_{I}}-\...
In quantum mechanics, tensor operators are defined through their commutation with the operators of spherical angular momentum components, \begin{aligned} \ [L_3,T(k,q)] &= \hbar q\ T(k,q), \\ [...
In Dirac's 'Principles of Quantum Mechanics' the definition of quantum Poisson bracket is $uv-vu=i\hbar[u,v]$ with $[q_r,p_s]=\delta_{rs}$. But in lots of other books the definition is $uv-vu=[u,v]$ ...