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1answer
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Fundamental Poisson Bracket under Canonical Transformation

We have $H(q,Q,t)$. There is unique solution $Q$ for $p_i = \frac{\partial H}{\partial q_i}$ and we have $P_i := - \frac{\partial H}{\partial Q_i}$. We want to proof from only this that the ...
3
votes
1answer
49 views

Fermionic Poisson bracket

I'd like to understand the Poisson bracket for fermions in classical field theory defined on a cylinder (with coordinates $(t,x)$, $x$ being the compact direction) and propagating on $T^n$ with ...
4
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0answers
159 views

What exactly is the relationship between the symplectic 2-form and the frequency of leaves of integrable systems in classical mechanics?

In classical mechanics we equip a differential manifold with a closed symplectic 2-form $\omega$. The symplectic leaves of integrable systems also have a unique frequency, in literature denoted ...
1
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1answer
98 views

Dirac bracket for the Majorana Lagrangian

Note: See update below. Consider the Majorana Lagrangian $$\mathcal{L}=-\psi ^{\mathrm{T}}\mathrm{i}% \gamma ^{0}\left( \gamma ^{\rho }\partial _{\rho }+m\right) \psi ,\tag{1}$$ where $% \psi \in ...
5
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1answer
231 views

Symplectic leaves, tori and Poisson manifolds

For classical systems we can define a configuration manifold, whose cotangent bundle is a momentum phase space equipped with a closed, non-degenerate 2-form. Upon the commutative algebra of smooth ...
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0answers
11 views

Poisson bracket in 4d phase space

In phase space determined by $(q_i,p_i)$ the Poisson bracket of two functions f and g is $$\{f,g\}=\sum_{i=1}^N\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial g}{\partial ...
2
votes
1answer
56 views

Poisson brackets in curved spacetime

The time evolution of any field $\phi$ is given in terms of the Poisson bracket with the Hamiltonian, $$ \frac{\partial\phi}{\partial t} = \{\phi, H\}. $$ How does this relation change in curved ...
8
votes
1answer
151 views

What is the physical interpretation of the Poisson bracket [duplicate]

Apologies if this is a really basic question, but what is the physical interpretation of the Poisson bracket in classical mechanics? In particular, how should one interpret the relation between the ...
0
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0answers
94 views

Integrability in classical mechanics

An integrable system in classical mechanics is defined by action-angle variables and closed loop trajectories in phase space. I have also heard that the flow lines of an integrable system are ...
0
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1answer
91 views

Simple explanation of first and second class constraints with an example

Can someone give a simple physical example of first class and second class constraints? I mean, if you were giving a classical mechanics lecture for undergraduates, how would you explain this concept ...
1
vote
2answers
147 views

Is a vector field not a vector quantity?

I'm trying to make sense of Poisson bracket relation $$\{L_i,A_k\}_{PB}~=~\epsilon_{ikl}A_l,\tag1$$ where $L_i$ is $i$th component of angular momentum, $A_k$ is $k$th component of an arbitrary ...
0
votes
1answer
56 views

Infinitesimal transformations and Poisson brackets [duplicate]

I want to understand how bracket operations in general are related to symmetry and infinitesimal transformations (in hindsight of quantumfieldtheory), so I calculated an example with a particle that ...
2
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0answers
76 views

Hodge dual and the Moyal bracket? Any link? [closed]

I have already asked this on the mathematics Stack exchange but I thought I'd try it here too! The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an ...
0
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1answer
48 views

If $(q,p)$ to $(Q,P)$ is a canonical transformation, then does this imply $(Q,P)$ to $(q,p)$ is also?

If $(q,p)$ to $(Q,P)$ is a canonical transformation, then does this imply $(Q,P)$ to $(q,p)$ is also, assuming Hamilton's equations hold for the coordinates $(q,p)$? This seems like it should be true ...
9
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1answer
596 views

Understanding Poisson brackets

In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
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0answers
58 views

Calculation of the Poisson bracket of a (Classical) Yang-Mills generator

This question might be too technical or minute, but I believe someone can give me the right advise. What I want to calculate is a Poisson bracket algebra of classical YM gauge generators, ...
5
votes
2answers
323 views

Poisson brackets of the Kepler Problem

For the hamiltonian of a particle of unit mass in a kepler potential: $$H = \frac{1}{2}\mathbf{p} \cdot \mathbf{p} - \frac{\mu}{r}$$ The angular momentum vector is given by: $\mathbf{L} = \mathbf{r} ...
2
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1answer
112 views

Landau's Problem - Poisson bracks of a spherical symmetry function and angular momuntum in z axis

In landau's Mechanics, there's a problem: I think, if the function has the property spherical symmetry, or: $\phi(r,p)=\phi(-r,-p)$ The form suggested by Landau follows this property, but I can't ...
2
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1answer
104 views

Poisson brackets for constrained system

Let's have some Hamiltonian which involves the set of first class constraints $\varphi$ and set of constraints $\kappa $, which play role of canonical conjugated momentums for $\varphi$,. They're ...
3
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1answer
123 views

Yang-Mills constraints and Poisson brackets

Let's have constraints for Yang-Mills theory: $$ \varphi_{a} = \partial_{i}\pi^{i}_{a} - f_{abc}\pi^{b}_{i}A^{c}_{i}. $$ I have read the statement that $$ \tag 1 [\varphi_{a}(\mathbf x), ...
8
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3answers
272 views

When do phase space functions' Poisson brackets inherit the Lie algebra structure of a symmetry?

I've seen several examples of phase space functions whose Poisson brackets (or Dirac brackets) have the same algebra as the Lie algebra of some symmetry. For example, for plain old particle motion in ...
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0answers
72 views

Why are Lagrangian subspaces called 'Lagrangian'?

I am wondering what the special role of Lagrangian subspaces (or submanifolds) are in mechanics. Do these subspaces have some sort of special property for which we have some sort of `Lagrangian ...
6
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1answer
314 views

Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
3
votes
1answer
219 views

Geometric mechanics - Symplecticity

I am just trying to wade through literature on classical mechanics and I really don't know where to start, everything is Fibre bundle this or manifold that, and doesn't really ease you in to the ...
4
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3answers
187 views

About constraints of the first class and electrodynamics

Consider a theory in the Hamiltonian formalism and assume that it has constraints between canonical variables $Q, \pi$. By the Dirac terminology, the set of constraints $F_{a}(Q, \pi) \approx 0$ of ...
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0answers
30 views

freedom of choice of 1-form in canonical representation of generic local field corresponds to gauge choice?

So it is a question in Gravitation Wheeler, Thorne and Misner 4.2 Exercise. Given F=$dp_{i}\wedge dq^{i}$. Using canonical transformation from p to $\bar{p}$ and q to $\bar{q}$, one gets ...
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0answers
49 views

How to show $ \epsilon_{iab}\epsilon_{jcd}(x_ap_d\lbrace x_c,p_b \rbrace+x_cp_b\lbrace x_a,p_d \rbrace) = x_ip_j-x_jp_i$ [closed]

If $ \lbrace f,g \rbrace $ is Poisson bracket and $\epsilon_{ijk}$ is Levi-Civita symbol, how to show that $$ \epsilon_{iab}\epsilon_{jcd}(x_ap_d\lbrace x_c,p_b \rbrace+x_cp_b\lbrace x_a,p_d ...
4
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0answers
863 views

Show that the Laplace-Runge-Lenz vector is conserved using poisson brackets

(I realise similar Phys.SE questions already exist but there is no answer with the Poisson bracket notation, I'll take this down if someone lets me know I should have commented in the existing ...
4
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1answer
177 views

A question about canonical transformation

I have posted this question in math.stackexchange before with no answer till now. It may be more suitable to post here. There is a problem in Arnold's Mathematical Methods of Classical Mechanics ...
2
votes
1answer
61 views

How exactly is the Poisson bracket of the modes of a classical string defined?

In the theory of a classical bosonic string, we have expressions like: $$ \{\alpha^\mu_m,\alpha^\nu_n \} = - i m \delta_{m,-n} \eta^{\mu \nu} $$ were $\alpha^\mu_n$ are the Fourier modes of the ...
4
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2answers
298 views

Hamilton formalism for Dirac spinors

Let's have the Dirac free lagrangian: $$ L = \bar {\Psi} (i\gamma^{\mu}\partial_{\mu} - m) \Psi . $$ I can rewrite it as $$ L = i\Psi^{\dagger}\partial_{0}\Psi - H_{d}, \quad H_{d} = ...
2
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0answers
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Questions about closed forms and cycles

I read the section closed forms and cycles in Arnold's Mathematical Methods of Classical Mechanics (page 196-200), but the problems in this section is too difficult to solve in the way following the ...
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0answers
81 views

Hamiltonian formalism in quantum electrodynamics

I need to compute $\frac{d}{dt}\hat{\mathbf P} = \frac{d}{dt}(\hat{\mathbf p} - q\hat{\mathbf A})$ for the solutions of $$ (i\gamma^{\mu}\partial_{\mu} + q\gamma^{\mu}A_{\mu} - m)\Psi = 0. $$ May I ...
6
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1answer
266 views

Physical intuition for deformation quantization of Poisson manifolds

First of all, I know almost nothing about physics. I was reading Kontsevich´s paper on Deformation quantization of Poisson manifolds, however I could not figure out what´s the intuition for such ...
0
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1answer
266 views

Calculate integral of motion condition with Poisson brackets

Problem statement: The Hamiltonian of a system is given by the formula: \begin{equation*} H = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2} + V(r,\theta). \end{equation*} Under what condition is ...
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2answers
518 views

Canonical equal time commutation relations in QED

I understand that to quantize the classical electromagnetic field one needs to impose commutation relations and express the field in terms of creation and annihilation operators. I notice that the ...
0
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0answers
132 views

Infinitesimal transformations and Poisson bracket for Dirac spinors

I apologize for the cumbersome calculations. Let's have $\Psi$, $i\Psi^{\dagger}$, which are canonical coordinate and impulse in space of solutions of Dirac equation. It can be showed that they have ...
5
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2answers
462 views

Heisenberg picture of QM as a result of Hamilton formalism

Consider the formula for the total time-derivative of a physical value in Poisson's formalism: $$\tag{1} \frac{dA}{dt} = -\{H, A\}_{P.B.} + \frac{\partial A}{\partial t}, $$ where $\{A, B\}_{P.B.}$ is ...
1
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2answers
437 views

Full time-derivative of a function and Schrodinger equation

From Hamiltonian formalism there is well known equation, $$ \frac{d F}{dt} = \frac{\partial F}{\partial t} + \{F, H\}_{PB}, $$ where $ \{H, F\}_{PB}$ is the Poisson bracket. After using Hamiltonian ...
1
vote
3answers
3k views

How to find out whether a transformation is a canonical transformation?

We had a couple of examples where we were supposed to calculate the Canonical Transformation (CT), but we never actually talked about a condition that decides whether a transformation is a canonical ...
3
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1answer
242 views

Two components of angular momentum conserved $\Rightarrow $ All three components are conserved?

I was wondering whether it is correct to say that if two components of the angular momentum are conserved, then all three Cartesian coordinates of the angular momentum are conserved? I would regard ...
1
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1answer
310 views

Why is $\{Q, P\} = 1$ for a canonical transformation?

Why is $\{Q, P\} = 1$ for a canonical transformation? Given $P(p,q)$ and $Q(p,q)$.
2
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0answers
264 views

The correspondence between Poisson bracket and Commutators in Quantum Mechanics

I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this. How can we generally show that ...
5
votes
3answers
2k views

Physical interpretation of Poisson bracket properties

In classical Hamiltonian mechanics evolution of any observable (scalar function on a manifold in hand) is given as $$\frac{dA}{dt} = \{A,H\}+\frac{\partial A}{\partial t}$$ So Poisson bracket is a ...
4
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2answers
326 views

The string Poisson bracket

Where does the factor $\frac{1}{T}$ ($T$ is the string tension) in this Poisson bracket come from? $$ \{X^{\mu}(\tau,\sigma),\dot{X}^{\nu}(\tau,\sigma')\} ~=~ ...
6
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2answers
248 views

Are Poisson brackets of second-class constraints independent of the canonical coordinates?

Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\{\chi_N(q,p),\chi_M(q,p)\}_P$ . Is ...
5
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1answer
559 views

Find the Hamiltonian given $\dot p$ and $\dot q$

I have these equations: $$\dot p=ap+bq,$$ $$\dot q=cp+dq,$$ and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$. To answer to the first ...
4
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1answer
434 views

Clarifications about Poisson brackets and Levi-Civita symbol

I need some clarifications about Poisson brackets. I know canonical brackets and the properties of Poisson Brackets and I also know something about Levi-Civita symbol (definition and basic ...
1
vote
1answer
587 views

Canonical transformation and Hamilton's equations

I was trying to prove, that for a transformation to be Canonical, one must have a relationship: $$ \left\{ Q_a,P_i \right\} = \delta_{ai} $$ Where $Q_a = Q_a(p_i,q_i)$ and $P_a = P_a(p_i,q_i)$. Now ...
1
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1answer
2k views

Poisson brackets and angular momentum

I'm trying to find $[M_i, M_j]$ Poisson brackets. $$\{M_i, M_j\}=\sum_l \left(\frac{\partial M_i}{\partial q_l}\frac{\partial M_j}{\partial p_l}-\frac{\partial M_i}{\partial p_l}\frac{\partial ...