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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1answer
79 views

How to use the Poisson Bracket to generate a finite Lorentz transformation?

I'm doing something very wrong or It seems to me that I can't generate a finite Lorentz transformation using the exponential of the infinitesimal Lorentz boosts. Let me define $L_{x;v}$ as the ...
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Derivation of Poisson bracket and commutator of position and conserve charge [closed]

How can I prove these two relations? Assuming $\mathbf{D}$ is defined as $$\mathbf{D}=\sum\frac{\partial\mathcal{L}}{\partial\dot{x}_i}\delta x_i-\mathcal{L}=\mathcal{H}t-\frac{1}{2}\mathbf{p}\cdot\...
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1answer
62 views

Understanding the Functional Poisson Bracket

In classical field theory (for a single field $\psi$) the dynamical variables are defined to be functions of the fields $\psi$, $\pi$, $\partial_{x_{i}}\psi$ and maybe $\mathbf{r}$, where $\pi$ is the ...
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1answer
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Time evolution of Galilean boost

I was introduced the generator of Galilean boost $K=mx-pt$. I was given an Hamiltonian with several particles: $H=\sum_i \frac{p_i^2}{2m_i}+V(|x_i-x_j|)$ where the potential only depends on the ...
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How to obtain commutation relations from symplectic potential?

I am studying the notes on susy qm in 1 dimension of David Skinner (http://www.damtp.cam.ac.uk/user/dbs26/SUSY.html) (which itself follows the mirror symmetry book by Vafa and Hori (relevant pp. 206 - ...
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Poisson Bracket $\{\delta_{ij}, g\}$ and partial derivative of Kronecker delta

I am currently working through Shankar's Princeiple of Quantum Mechanics Exercise 2.8.2 is to verify that the infinitesimal transformation generated by any dynamical variable g is a canonical ...
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1answer
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Poisson Bracket of $\{Q, P\}$ in the original coordinate $(q, p)$

For simplicity, I use $(q,p)$ and $(Q,P)$ instead of $(q_i,p_i)$ and $(Q_i,P_i)$. I know that we should get $\{Q, P\} = 1$ for a canonical transformation $(q,p)\rightarrow(Q,P)$. But we also know from ...
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What is the “secret ” behind canonical quantization?

The way (and perhaps most students around the world) I was taught QM is very weird. There is no intuitive explanations or understanding. Instead we were given a recipe on how to quantize a classical ...
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Physics that calls for deeply nested Lie/Poisson brackets

I've been scouring physics for non-associative situations, particularly where study of quasigroups and loops might come in handy (they always seem to be left out). The poisson and lie brackets form a ...
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Quantization of an $\mathcal{c}$-algebra

I don't know if this is a valid question to ask, but I am wondering about the following: We are given a set of $\mathcal{c}$-number Lie-Brackets $$ [q_i,q_j] = 0= [p_i,p_j] \\ [q_i,p_j] = c_{ij}, $$ ...
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Addition of a constant to the operator due to quantization

Groenewold in his book On the Principles of Elementary Quantum Mechanics (1946, Springer Netherlands) page 45, maps the canonical momentum $p^2$ in the classical phase space to a general canonical ...
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Quantization of $c$-number Dirac-Bracket

I have a question concerning the quantization of phase-space variables $(q_1, q_2, q_3, p_1, p_2, p_3)$ with the Hamiltonian $$ H = \frac{3}{2}(p_1^2+p_2^2 +p_3^2) $$ and the following non-commuting ...
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1answer
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Is The Seiberg-Witten Map Unique?

From my understanding the Seiberg-Witten map is a way to convert a non-commutative field theory into a commutative field theory. For example for the commutative relation between positions $[x, y]=i \...
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Analogous structure of Diffusion and Schrödinger equation and definition of flux?

I came across some analogous structure of diffusion and the quantum mechanical particle (Schrödinger eq.). I have seen that there have been similar questions asked, but the (probablitily flux and the ...
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Validity of Canonical Quantization

I was studying about what does it mean canonical quantization treatment. But now I have the next question. Why if we establish canonical the commutation relations $$\left[q,p\right]=i\hbar,\quad \left[...
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How are the two definitions of Canonical Transformations related/equivalent? [duplicate]

I am aware of two definitions of canonical transformations which I state below. Definition $1$ We go from old set of $\{q_i,p_i,t\}$ of $2n$ phase space variables to a new set $\{Q(q_i,p_i,t),P(q_i,...
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Poisson bracket properties for tensor densities

I am doing some constraint analysis in an extended theory of gravity, and I am confused about Poisson brackets. The standard PB relations are for example $\{ab,c\} = a\{b,c\} + \{a,c\}b$ etc. But I am ...
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Change in Hamiltonian under infinitesimal canonical transformation [duplicate]

Consider an infinitesimal canonical transformation from the (symplectic) coordinates $z$ with Hamiltonian $H(z, t)$, to the coordinates $$ Z = z + \epsilon J \frac{\partial G}{\partial z} $$ with the ...
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2answers
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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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1answer
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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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1answer
86 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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2answers
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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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1answer
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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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135 views

Poisson brackets of three dimensional angular momentum and its Lie algebra

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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2answers
120 views

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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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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1answer
90 views

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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1answer
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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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1answer
75 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
75 views

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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1answer
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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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1answer
98 views

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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1answer
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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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1answer
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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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1answer
177 views

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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1answer
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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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1answer
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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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1answer
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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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1answer
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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 ...
2
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1answer
66 views

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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2answers
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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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1answer
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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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