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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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Proving Poisson bracket relations $\{\phi, P^r\}=\Pi^r$ in Ticciati's “QFT for Mathematicians”

Let $\phi$ be a scalar field, and $\Pi$ be the conjugate momentum of $\phi$. Let $\cal L=\cal L(\phi, \partial_\mu \phi)$ be the Lagrangian density. Define the stress-energy tensor as $$ T^{\mu\nu}=\...
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Is there always a canonical transformation such that the new Hamiltonian only depends on the new momenta?

Given the Hamiltonian $H(x,p)$ of a system. Is there always a coordinate transformation such that the new Hamiltonian is $K(x',p')=K(p')$?
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Couple of non-interacting, non-integrable Hamiltonian systems

I have two Hamiltonians, $H(p_1, q_1, \dots, p_n, q_n)$ and $H'(p_{n+1}, q_{n+1}, \dots, p_m, q_m)$. They are non-interacting, indeed they have different variables. Moreover, I know that they are both ...
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A paradox about canonical transform preserving Poisson bracket?

Let $q,p$ denote the position and momentum. Consider a transform generated by $g$: $q' = q + \epsilon \{q,g\}---(1a)$ $p' = p + \epsilon \{p,g\}---(1b)$ Then: $\{q',p'\} = \{q,p\}+o(\epsilon^2)+\...
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Symplectic Manifolds in General Relativity for Integrable Systems

To solve the geodesic equations for a specific metric in General Relativity I can find conserved quantities $F = \xi_{\mu}\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\lambda}$ along geodesics by using Killing ...
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Generating function depending on $q$, $p$, $Q$ and $P$

If I have a generating function, say, $$G(q,p,P,Q)= qp - e^Q e^P\tag{1}$$ what are the equations that give me the transformations $Q=Q(p,q)$ and $P=P(q,p)$? I have only seen generating functions ...
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Integration of Poisson brackets by integration by parts [closed]

In the context of Statistical Mechanics I have to show that the following integral is zero: $$\int \sum_{i=1}^{3N}(\frac{\partial O}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial O}{\...
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Why is a single function sufficient to specify a canonical transformation?

Spivak argues at page 577 in his book Physics for Mathematicians: What are the $2n$ relations he is talking about?
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Are symmetries necessarily canonical transformations?

A canonical transformation is defined as a transformation such that afterwards Hamilton's equations still hold. It can then be shown that this requirement implies that canonical transformations are ...
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How much information about a quantum operator is determined by its Poisson bracket Lie algebra?

Hamiltonian quantum mechanics is often built using many ideas from Hamiltonian classical mechanics like the Poisson bracket to determine the commutator between quantum operators, which is appropriate ...
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How does a vanishing $[x, p]$ work with the group theoretical definition of $p \propto \frac{\partial}{\partial x}$?

Thought about this while I was looking at some stuff on quantum-classical correspondence and where precisely the difference between quantum and classical comes from. Usually it's said that the key/...
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Fluid mechanics, symplectic structure, the Hamiltonian, and vorticity

Consider an inviscid irrotational fluid in two dimensions. There are some explicit connections with symplectic geometry that I do not understand. I am not well versed in the later topic, so please ...
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Canonical coordinate

Sorry for my broken English. I'm a physics undergrad and quite poor at math. While reading a mechanics textbook, I've found something I cannot understand. There are coordinates, $(q,p,t)$ $\...
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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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Poisson Brackets in inhomogenous magnetic field [closed]

This question came in my classical mechanics paper and I still can’t solve it. A particle of mass $m$ and electric charge $e$ is moving under the influence of an inhomogeneous magnetic field $\vec{...
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Lapse and shift inside or outside the Poisson bracket?

For general relativity in the 3+1 ADM formulation, one has $H=\int dx [N{\cal H}+N^a{\cal H}_a]$ with $N$ and $N^a$ the lapse and shift which are undetermined Lagrange multipliers. The dynamical ...
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Poisson Brackets and Angular Momentum using Poisson Bracket algebra

I've been trying to prove the identity for $\left\{L_i,L_j\right\}$ using only the algebra for the Poisson Brackets (of course I could do it by the definition with the derivatives and such, but where'...
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Problem on deriving canonical transformation condition

I'm trying to compute how a canonical transformation should be, given that preserve the symplectic form and trying to recover the condition on the Poisson Bracket. I then start with $$\omega=\stackrel{...
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Hamiltonian Structure of Chern Simons Electrodynamics

I am reading the review paper "Aspects of Chern-Simons Theory" by Gerald Dunne https://arxiv.org/abs/hep-th/9902115 Starting from p. 17, Dunne works on the Hamiltonian structure of the CS ...
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Poisson Bracket in General Relativity and tensor weight

I'm a bit confused about the tensor density weight of Poisson brackets in general relativity and their covariance. It's perhaps related to being unclear as to what happens when I integrate a scalar ...
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Relationship between complex analysis and Hamilton's canonical equation

I recently came across a mathematical field called complex analysis. There was an important equation called Cauchy-Riemann equation. When I saw it at first, I recalled a book's sentence stating ...
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Dirac bracket for the Madelung (polar) form of the Schrodinger field

I'm having an issue with obtaining the Dirac bracket in the Madelung (polar) representation of the Schrödinger field: \begin{equation} \Psi=\sqrt{\rho}e^{i\theta/\hbar}. \label{eq:...
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A confusing point of the Hamiltonian for a particle interacting with electromagnetic fields

In non-relativistic quantum theory the Hamiltonian for a particle interacting with electromagnetic fields is $$H=\frac{(\mathbf{p}-\mathbf{A}*e/c)^2}{2m}+e\phi+\int\,d^3x \frac{\mathbf{E^2}+\mathbf{B^...
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Is there a physical interpretation of symplectic manifolds which are not cotangent bundles?

The inspiration for symplectic geometry was from Hamiltonian mechanics. However, I am wondering how close the ties are between arbitrary symplectic manifolds and real physical systems. In ...
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Physical interpretation of differences between classical and quantum ensemble dynamics

The Groenewold-Moyal (phase space) picture of quantum mechanics describes the evolution of a probability density corresponding to a wavefunction that evolves as described by Schrödinger's equation. ...
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Problem with the regularity condition for a constraint

So, I'm considering Lagrangian: $$L=\frac{1} {2}e^{q_1}\dot{q}_2^2. $$ I obtain the primary constraint $\phi=p_1=0$. The canonical Hamiltonian is $H_c=\frac{1} {2}p_2^2e^{-q_1} $, and the total ...
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Poisson Brackets And Angular Momentum Components

Related: Poisson brackets of angular momentum When Poisson Brackets are taught as part of an Analytical Mechanics courses, examples are commonly shown which anticipate analogue results in QM. One ...
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One dimensional system a Hamiltonian system?

I have the following equation of motion: $$ \dot x = \beta x y $$ with $y=1-x$. I would like to see if it is Hamiltonian or not. Due to it being one dimensional, I think it should be locally ...
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Determining if constants of motion are independent

Say, in Hamiltonian mechanics, we know two constants of motion, $A$ and $B$. It could be proven that the quantity $[A,B]$ is also a constant of motion, where $[A,B]$ denotes the Poisson brackets of $...
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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}}-\...
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1answer
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Tensor operator analogy in classical physics

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), \\ [...
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Two different definition of quantum Poisson bracket, which one to use?

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 $uv-vu=[u,v]$ ...
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Proof of constructing Action-Angle Coordinates on Hamiltonian System

By Liouville-Arnold Theorem, we know we can construct action-angle coordinates such that the Hamiltonian system, when described in these coordinates, will have a form that is integrable by quadratures....
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How to obtain Hamiltonian formalism and phase space for Lagrangian with second-derivatives?

This is a special case of this question of mine, which, I think, might have drawn little attention because it was too general. In this question, I would like to consider a specific case. Take a ...
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Symplectic form on covariant phase space

Usually, the phase space of a physical system is defined as the cotangent bundle of the configuration space at some fixed time slice $t = t_0$, conveniently co-ordinatized by $\{q^a, p_a\}$ where $$ ...
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Canonical transformation of spacetime or under field theory formalism

Is there a standard way to define Canonical Transformations in the case of field theory or spacetime? I saw that under ADM formalism it is possible to define a generating function: $$ G(t)= \sum_i ...
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Is a canonical transformation equivalent to a transformation that preserves volume and orientation?

We have seen the reverse statement: Lioville's Theorem states that canonical transformations preserve volume (and orientation as well). Is the reverse true? If I demand a map from the phase space to ...
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Can any symplectomorphism be called a canonical transformation?

I just want to make sure I am thinking clearly about canonical coordinates and transformations in Hamiltonian mechanics. Suppose we have a Hamiltonian system $(M, \omega, H)$ — where $M$ is the ...
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Hamilton equations from Poisson bracket's formulation

Referring to Wikipedia we have that the equation of motion for a $f(q, p, t)$ comes from the formula \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t} f(p, q, t) = \frac{\partial f}{\partial q} \frac{\...
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Does Schroedinger equation depend on the sign of Poisson bracket?

Let's consider Poisson bracket $$\left\{ A,B\right\} =\alpha_{p} \left( \frac{\partial A}{\partial p_{k}}\frac{\partial B}{\partial q^{k}}-\frac{% \partial A}{\partial q^{k}}\frac{\partial B}{\...