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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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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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Expansion to show $g$ is conserved if $H$ is invariant

On Shankar QM page 99 it says that If $H$ is invariant under the following infinitesimal transformation $$q_i\rightarrow\bar{q_i}=q_i +\epsilon\frac{\partial{g}}{\partial{p_i}}$$ $$p_i\...
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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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How to determine whether two variables are canonical or not?

I'm currently reading about Ashtekar's variables, and found out that the new variables, $A$ and $E$, are canonical, so they satisfy the following poisson bracket \begin{eqnarray} \left\{ A^i_a(x), A^...
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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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1answer
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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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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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Conservation of Angular Momentum for Real Scalar Field

This is my first question in Stack, glad to join the community! Here it goes: I'm trying to prove that angular momentum for a real scalar field, satisfying Klein-Gordon equation, is conserved via ...
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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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1answer
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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}{\...
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Poisson brackets' property as necessary and sufficient condition for transformation to be canonical?

I read (Landau, Lifshitz: Mechanics) and then I want to know if conditions (45.10) are sufficient for transformation $p,q \to P,Q$ to be canonical (obviously, they are necessary).
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Why are Hamiltonian's used with non-Hamiltonian brackets to define equations of motion?

I am reading Non-Hamiltonian equilibrium statistical mechanics and I am confused about how they are using Hamiltonians. In Section II, they introduce the "non-Hamiltonian bracket", \begin{align} \{a, ...
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Hamiltonian Dynamics for a rigid body, torque free

As we know that in classical mechanics, $$\frac{d}{dt}f=\{f,H\}+\frac{\partial}{\partial t}f$$ where the second term appears when $f$ is explicitly time dependent. Here is my question: When we ...
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Is it possible to obtain an analogous of Liouville's Theorem in terms of the Configuration Space?

Is it possible to obtain an analogous of Liouville's Theorem in terms of the Configuration Space (or the tangent space thereof)? If not, why is this result obtainable only in the Hamiltonian ...
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Proof of a property of the Poisson bracket

I have seen written in many courses of statistical mechanics that, for two functions of the general coordinates and momenta $f(q,p)$ and $g(q,p)$ to satisfy $$ \{f,g\}=0 \tag{1} $$ in a 2D phase ...
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Sufficient conditions for a mapping to be canonical in Hamiltonian Mechanics

My professor mentioned: A simple way of testing whether a mapping $(q,p)$ to $(Q,P)$ is canonical is by examining: $$P · dQ − p · dq$$ and if it equals to $dA$ (a differential) then it is canonical. ...
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Is there no coordinate conjugate to $L_x$?

The conjugate momentum corresponding to $\phi$ (azimuthal angle in sp. polar coordinate) is $L_z$ (sometimes written $L_\phi$) which is frequently used in quantum mechanics. Why is there no coordinate ...
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How does a Legendre transformation $H\to L$ work in non-canonical coordinates?

Let $H(z)$ be a Hamiltonian and $\omega_{ij}$ the symplectic form on the phase-space and $\omega^{ij}$ its inverse $\omega_{ij} \omega^{jk} = \delta^k_i$. We know that the Hamilton's equations are ...
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What is the commutator of a Poisson bracket and the covariant derivative?

Consider a classical vector field $V^\mu$ on a curved background. We make a 3+1 split of coordinates into $t,x^i$, where $x^i$ are coordinates on spatial hypersurfaces $\Sigma$ and $t$ the parameter ...
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The geometrical interpretation of the Poisson bracket

"Hamiltonian mechanics is geometry in phase spase." The Poisson bracket arises naturally in Hamiltonian mechanics, and since this theory has an elegant geometric interpretation, I'm interested in ...
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Time-dependent Canonical transformation

Suppose that the Hamiltonian of the mechanical system under analysis depends on two complex conjugate variables $a$ and $a^*$, so we have: $$ H=H\left(a,a^*\right) $$ Hamilton equations read $$ ...
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1answer
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Deriving the commutation relations in Maxwell-Chern-Simons theory

I am learning quantization of MCS theory. $$L_{MCS}=-\frac{1}{4}F^{\mu \nu}F_{\mu\nu}+\frac{g}{2} \epsilon^{\mu \nu \rho}A_\mu\partial_\nu A_\rho$$ I have reached the commutation relation $$[A_i(\...
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1answer
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In Hamiltonian field theory, do spatial derivatives commute with Poisson brackets?

I'm going back over some of my old notes for a current project, and I'm trying to figure out if I made an error or if I once knew something that I've now forgotten. Consider a local field theory ...
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Why is transverse delta distribution needed for Poisson brackets of classical EM field components?

During reading a German lecture notes (Quanten Optik by Dirk–Gunnar Welsch) about quantization of the EM field, I stumbled over a statement I cannot reproduce in detail: Generally, it is argued, that ...
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Expected value of commutator using path-integral

Consider a real scalar field theory in finite temeperature. According to the book by Kapusta and Gale, Finite-Temperature Field Theory, its retarded Green's function is given by $$iD^R(x,x') = Tr\{\...
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2answers
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Whence the $i$ in QM Poisson bracket definition?

On p. 87 of Dirac's Quantum Mechanics he introduces the quantum analog of the classical Poisson bracket$^1$ $$ [u,v]~=~\sum_r \left( \frac{\partial u}{\partial q_r}\frac{\partial u}{\partial p_r}- \...
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Poisson brackets

I want to find out,for which constants (the greek letters) is this transformation canonical: $$Q^1 = \alpha q^1, Q^2 = q^2 + \beta (q^1)^3, P_1 = p_1 + \gamma (q^1)^2p_2\text{ and }P_2 = \delta p_2~?$$...