# 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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### Lie algebra of Hamiltonian functions

Could anyone explain to me how to obtain the Corollaries 2 and 3 in the following paragraph taken by Arnold's book? The Lie algebra of hamiltonian functions The hamiltonian vector fields on a ...
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### In equation (20) from lecture 10 in Leonard Susskind’s ‘Classical Mechanics’, why is there a summation involved?

Here is the equation $$\{x_i,L_j\}=\sum\limits_{k}ϵ_{ijk} x_k.$$ Is this equation generalised for any number of dimensions? In which case, would the following example be correct assuming 4 dimensions? ...
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### Is it possible to understand in simple terms what a Symplectic Structure is?

I would like to understand what a Symplectic Structure is, and its implications in Classical Mechanics (Phase Space), but in pre-grade terms (If that could be possible). I have not taken any ...
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### Derivation of self-dual gravity formulae

I am trying to read and understand this paper by Monteiro, Stark-Muchao, and Wikeley about self-dual yang-mills and self-dual gravity. In the introduction to this paper, they review a way to ...
1 vote
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### Canonical transformations in the covariant phase space formalism

As the title says, I'm looking for an explanation on how to apply canonical transformations when using the covariant phase space formalism. I'm familiar with the topic, but I haven't found a good ...
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### How to interpret Poisson bracket of fields in terms of causality?

In quantum field theory, the fact that space-like separated observables commute, i.e. $[\hat {\phi (x)}, \hat{\phi(y)}]=0$, is taken as the test for causality. The equivalent statement for classical ...
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### Proof of Liouville's Theorem

The Wikipedia article on Canonical Transformations has a section on Liouville's Theorem. It makes the following argument: $$J=\frac{\partial(\mathbf{Q},\mathbf{P})}{\partial(\mathbf{q},\mathbf{p})}$$ ...
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### A problem understanding primary constraints meaning

I have some problems understanding the meaning of a function that vanishes weakly. As far as I can understand, when somebody writes that a function $F$ in the phase space vanishes weakly, that means ...
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### Is there any classical correspondence of the Jacobi identity? [closed]

In QM, the commutator were closely related to the poison bracket, so much so that to promote the classical operators to the quantum operators were often associated as \{A,B\} \text{(...
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### How to show that the Hamiltonian $H$ is invariant under flow generated by $F$?
I know usually if I have a transformation of phase space $Q(p,q), P(p,q)$ it is defined to be canonical if and only if its Jacobi matrix is part of the Symplectic Group or equivalently \$\{Q^{i}, P_j \}...