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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Liouville measure conservation and smoothness of the symplectic structure
Suppose we have hamiltonian $H$ of some system. Suppose we fix the coordinate $q$ and momentum $p$, which are coordinates in the phase volume of the system. In general, they are not canonical. This ...
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Example of a transformation that is not canonical
Can someone please give me an example of a transformation that is not a canonical transformation?
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Calculation of the standard symplectic space matrix
I am learning there is an important connection between Hamiltonian formalisms and Symplectic Geometry. It seems like the Newtonian Mechanics are described on what is called the standard symplectic ...
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Quantization from Poisson brackets
It is known that given a symplectic manifold, one can define different Poisson brackets. I am trying to see whether in classical sense ($\hbar=0$) given two different Poisson brackets (i.e. a ...
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How does one know if two variables are conjugate pairs?
First of all, I am having a hard time finding any good definition of what a conjugate pair actually is in terms of physical variables, and yet I have read a number of different things which use the ...
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Norm of Classical (Poissonian) Hamiltonian Operator
In the Poissonian formulation of classical mechanics, one finds that the time evolution of the phase space vector $\eta = (q_1,q_2\cdots q_n ; p_1, p_2\cdots p_n)^T$ can be put in terms of the ...
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I cannot verify if given transformation is canonical (CT) [closed]
I'm going on a list of exercises and there's 6 days that I can't figure to this.
Prove that given transformation is canonical:
$$\Large Q=\frac{p}{mw}\sin(\frac{mwq}{P}); \qquad\Large P=p\cos(\...
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Poisson brackets among derivatives in the constraints of the Hamiltonian formulation of General Relativity
The issue I have is from the Hamiltonian formulation of General Relativity (GR) and concerns calculation of certain Poisson brackets. In this formulation, we have the 3-dimensional metric $h_{ij}(x)$ ...
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Significance of symplectic form in classical field theory
I'm trying to understand the significance of construction presented to me in field theory class. Let me first briefly describe it and then ask questions.
Given two solutions $\phi_1$, $\phi_2$ of the ...
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Hamiltonian from a differential equation
In my differential equations course an example is given from the Lotka-Volterra system of equations:
$$ x'=x-xy$$
$$y'=-\gamma y+xy.\tag{1}$$
This is then transformed by the substitution: $q=\ln x, ...
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Quantum systems without a classical analogue? [closed]
I am now reading the quantum mechanics textbook by Dirac (chap. 4, $\S21$, p. 88).
He says that his quantization procedure does not include all possible systems in quantum mechanics and there are ...
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Non-holonomic constraints in Dirac-Bergmann theory
The Dirac-Bergmann algorithm effectively isolates the physical degrees of freedom of a system, by changing from Poisson brackets $\{\cdot,\cdot\}_\mathrm{PB}$ to Dirac brackets $\{\cdot,\cdot\}_\...
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How do I obtain the SUSY Transformations from Poisson Brackets?
In Friedman's and Van Proyen's Supergravity textbook it is explained how one can get the supersymmetry transformations using the conserved currents. Specifically this is in section 6 where we are ...
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Motivation for covariant phase space
The covariant phase space idea, in one sentence, is that there is a natural symplectic structure on the space of the classical trajectories of a system and that the usual $(q,p)$ coordinates just ...
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How is the Poisson bracket $\{\mathbf{c},\mathbf{l}\cdot\hat{n}\}=(\hat{n}\times \mathbf{c})$, for constant $\mathbf{c}$, and not zero?
The Poissonian formulation of mechanics tells us that for a generating function $g(q,p,t)$, the Poisson bracket of some function/variable $f(q,p,t)$ with the generating function corresponds with an ...
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Why are the integrability conditions necessary and sufficient for the existence of a canonical transformation's generating function?
Consider a canonical transformation $(p,q) \rightarrow (P,Q)$ under a generating function $F$. The condition for form invariance of Hamiltonian equations of motion looks like :
$$\sum_{s}P_s\dot{Q_s} ...
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Canonical Transformation [duplicate]
How can I prove that the following transformation is canonical:
$\begin{cases}\overline{q}_i=\dfrac{q_i}{Q} \\ \overline{p}_i=Qp_i-2Pq_i
\end{cases},\ i\in\overline{1,n}$
where $Q=\sum_{i=1}^n q^2_i$...
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Find the error: If $L_x$ and $L_y$ are zero, then $L_z$ is conserved
From Goldstein's Classical Mechanics (2nd ed.), problem 38 of chapter 9 basically says the following:
It's been shown that the Poisson bracket of two constants of the motion is also a constant of the ...
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Clever way to show a property of Lie transformation
Given the following Lie transformation:
$$
\exp(\lbrace H, \cdot \rbrace):=\sum_{n=0}^{\infty} \frac{(\lbrace H, \cdot \rbrace)^n}{n!}
$$
and apply it to a Poisson Bracket $\lbrace g_1, g_2 \...
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Field theory: equivalence between Hamiltonian and Lagrangian formulation
Let $\mathscr{B}$ be a space of physics we have and $\mathscr{T}$ be the duration.
Let $\mathscr{L}$ be a lagrangian density of the field such that the action is a functional of $\phi:\mathbb{R}^4\...
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How to impose canonical commutation relations when quantising a field
I believe this is a simple question, however I cannot find it explained anywhere what the term:
"Impose canonical commutation relations" means.
If I have a classical equation, and I wish to quantise ...
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A mysterious conserved quantity for a central potential
In teaching a course in classical mechanics and I have come across (from my predecessor) a to me mysterious conserved quantity.
We are considering a gravitational (or electric) potential with the ...
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How the Poisson bracket transform when we change coordinates?
I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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What are the implications of integrating the Poisson bracket?
Reading Ref. 1 I admit I am a little lost in some places. I was hoping that someone in this area could explain the basic premise of integrating the Lie bracket and further is it connected to nlab's ...
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Full time-derivative, Poisson brackets and Hamilton's equations (classical mechanics)
While studying Poisson brackets in classical mechanics and the derivation of $\dot{q_j}=\{q_j,H\}$ and $\dot{p_j}=\{p_j,H\}$ form of Hamilton's equations I encountered a surpsing identity, which led ...
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Canonical Commutation Relations in arbitrary Canonical Coordinates
If one were to formulate quantum mechanics in an arbitrary canonical coordinate system, does he impose canonical commutation relations using Dirac's recipe?
$$[\hat{Q}_i,\hat{P}_j]~=~i\hbar~\{q_i,p_j\...
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Landau/Lifschitz's proof of Jacobi's identity
The Poisson brackets of two quantities is defined as
$$[f,g]=\sum_k \Big( \frac{\partial f}{\partial p_k}\frac{\partial g}{ \partial q_k}- \frac{\partial f}{ \partial q_k}\frac{\partial g}{\partial ...
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Dirac bracket for a constrained particle
I am trying to work through a simple example of how to use the Dirac bracket from the following paper. In particular section 4 where the authors consider a constrained particle with the following ...
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Conservation of angular momentum poisson brackets vs newtonian mechanics [closed]
So I have the following system. For a mass falling due to gravity the given Hamiltonian is
$$
H = \frac{1}{2m}\left( P^{2}_{x} + P^{2}_{y} \right) + mgy
$$
In Cartesian coordinates then,
$$
x = v_o ...
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Deriving the Poisson bracket relation of the Ashtekar variables
I'm trying to figure out how to calculate the orthogonality of Ashtekar variables with respect to the ADM hypersurface metric and conjugate momentum.
$$\{{A_a}^i(x), {E^b}_j(y)\} = 8 \pi \beta \delta^...
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Is there any relation between Poisson Brackets and the Jacobian Matrix?
The Poisson brackets for $u,v$ can be written as,
$$
\frac{\partial u}{\partial q} \frac{\partial v}{\partial p} - \frac{\partial u}{\partial p}\frac{\partial v}{\partial q}.
$$
We can write this ...
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Does the poisson bracket $\{f,g\}$ have any meaning if neither of $f$ or $g$ is the system's Hamiltonian?
Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...
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Poisson brackets and magnetic field [closed]
I'm a maths student trying to teach myself some physics so sorry if I'm missing something simple here. I think the main problem is lack of experience with the Levi-Cevita symbol.
We have a particle ...
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Why closed in the definition of a symplectic structure?
Why do we want the 2-form $\omega $ to be closed? What if it is not?
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For an infinitesimal transformation in phase space, what functions are allowed for this to be a canonical transformation?
Consider an infinitesimal transformation:
$$(q_{i},p_{j}) \quad\longrightarrow \quad(Q_{i},P_{j}) ~=~ \left(q_{i} + \alpha F_{i}(q,p),~p_{j} + \alpha E_{j}(q,p)\right) $$
where $α$ is considered to ...
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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 prove from only this that the ...
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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 ...
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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 $\...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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Calculation of the Poisson bracket of a (Classical) Yang-Mills generator [duplicate]
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,
\begin{...
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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} ...
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