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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Jacobian rules with canonical transformations

If we consider a canonical transformation from $(q,p)$ to $(Q,P)$, it is stated in several sources that by Jacobian rules, $$ \frac{\partial(Q,P)}{\partial(q,p)} = \frac{\partial(Q,P)/\partial(q,P)}{...
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Non-hamiltonian systems which evolve into hamiltonian by change of coordinates

I am very new to the subject, so please forgive my very naïf question. I learned that there are some non-hamiltonian systems which can become hamiltonian, just by a change of coordinates. I was given ...
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Geometric Intuition for the Moyal Product

I've recently been reading into deformation quantization as another formulation of quantum mechanics. I have focused on understanding the Moyal product in particular, as it contains the seeds for the ...
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62 views

Why/Do Hamilton's equations Hold with Complex Variables?

I am investigating the problem of taking a hamiltonian of bilinear terms, and converting them into a bunch of uncoupled oscillators, such as in a periodic lattice. To do this, you have to introduce ...
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Definition of weakly vanishing

In literature concerning dirac constraints, often the word weakly vanishing is used say for example, a function $f$ is called first class if its poisson brackets with all the constraints $\phi_i$ ...
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Confusion regarding properties of Poisson Brackets

I have just started learning about Poisson Brackets, and came across the following property $$\{q_i,q_j\}=0$$ And $$\{p_i,p_j\}=0.$$ Where $p$ and $q$ are respectively the momentum and position ...
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Poisson bracket of momentum constraints in general relativity

I wish to compute the Poisson bracket of the momentum constraints in general relativity. Unfortunately, I am not able to do it correctly and the answer I am getting is not a linear combination of the ...
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53 views

Poisson bracket of Hamiltonian with Hamiltonian always vanishes

Since Poisson bracket of Hamiltonian with Hamiltonian always vanishes then in case of explicit time dependence of Hamiltonian, how does Poisson bracket gives correct result?
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Poisson brackets of three dimensional angular momentum and its Lie lagebra

I've recently noticed that the Poisson brackets of the three dimensional angular momentum $$\{L_i,L_j\}$$ in classical mechanics follow the same commutator relations as the standard basis of the Lie ...
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Generating function for canonical transformation

Short version: I've been reading through some notes on integrable systems/Hamiltonian dynamics, and am stuck on a problem: Show that the coordinate transformation derived via the generating function ...
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Possible relation between the canonical variables and Casimir operator

I studied the Goldstein book on the classical mechanics. On page 411 of its third edition, it is stated: Recall from the fundamental Poisson brackets, Eqs. (9.69), that the Poisson bracket of any ...
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CCRs and Classical equivalent formulae [duplicate]

I have managed to show that for the one-dimensional classical system (with two dimensional phase space) we have $$\left\{ q^{a}p^{b},q^{c}p^{d}\right\} =\left(ad-bc\right)q^{a+c-1}p^{b+d-1}$$ Where ...
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Connection between Poisson bracket and Anti-commutator?

Canonical quantization promotes Poisson brackets in classical mechanics to commutators in quantum mechanics. Is there any classical counterpart similar to the Poisson bracket for the anticommutator?
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Poisson Bracket of a Quantity Involving a Differential

I am working through Warren Siegel's "Fields" and have come across the following exercise on p. 58 involving an action measure and a symmetry generator: Exercise IA4.1. For general ...
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66 views

What is canonical momentum in rigid body systems?

What is the canonical momentum in rigid body systems? It appears to me that it can't be the angular momentum as these have non-vanishing Poisson brackets, but any canonical coordinates have $\left\{...
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1answer
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What is the symplectic form for rigid body systems?

I was pondering rigid body mechanics and the non-vanishing Poisson brackets $\left\{J_x,J_y\right\}=J_z$ etc. However in arbitrary coordinates $q^i$ with conjugate momenta $p_i$ we are supposed to ...
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Given Hamilton's Principal Function, $S(\alpha, \beta, t)$, prove that $p$ and $q$ are canonical variables

I've been self teaching using Hamill's Student's Guide to Lagrangians and Hamiltonians. (I know it's a terrible text to work from, but I'm almost done!) I'm stuck on problem 6.5 that basically asks, ...
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On- & off-shell conserved charges/constants of motion

I am trying to understand how conserved charges generate symmetry transformations via the Poisson bracket, but I am missing something in one part of the derivation. The part I am struggling with is ...
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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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Lax Pairs In Integrability

I am working through Dr. Beiserts notes (https://people.phys.ethz.ch/~nbeisert/lectures/IntHS16-Notes.pdf) and have difficulty obtaining the second step in (2.9): $$\{{\rm tr}L^{k},{\rm tr}L^{\ell}\} ...
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112 views

Time Dependence on Landau & Lifshitz's Proof of Poisson's Bracket Canonical Invariance

I'm reading Landau & Lifshitz's Mechanics and, at a certain point when discussing canonical transformations, they prove that Poisson brackets are canonical invariants. The proof starts with ...
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Dirac bracket and Poisson bracket, asymptotic symmetry

I am reading the paper arXiv:9906126. https://arxiv.org/abs/gr-qc/9906126 on the symmetry algebra at horizon (see also well known work done by Brown and Henneaux about the asymptotic algebra of AdS$...
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Canonical Transformations that are Complex

I'm self studying through a book that has the following question. The book gives the answer, but I'm trying to understand why: Under what condition is the following transformation NOT canonical? $$Q =...
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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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Angular momentum and rotation symmetry

In my book, it is written that for any vector $\mathbf{v}$, we have $$\{\mathbf{v},\mathbf{L}\cdot \mathbf{n}\}=\mathbf{n}\times\mathbf{v}.\tag{1}$$ For me it is absurd... For example, if we take $\...
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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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70 views

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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199 views

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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511 views

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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1answer
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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 ...