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# Questions tagged [phase-space]

A notional even-dimensional space representing all relevant states of a dynamical system; it normally consists of all components of position and momentum/velocity involved in that unique specification. Use for both classical and quantum physics.

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### Which transformations are canonical?

Which transformations are canonical? Why do canonical transformations preserve the measure of integration in phase space?
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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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### Topology of phase space

Context: From Liouville's integrability theorem we know that: If a system with $n$ degrees of freedom exhibits at least $n$ globally defined integrals of motion (i.e. first integrals), where all ...
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### Is Liouville's equation an axiom of classical statistical mechanics?

Suppose we have a classical statistical problem with canonical coordinates $\vec{q} = (q_1, q_2, \dots, q_n)$ and $\vec{p} = (p_1, p_2, \dots, p_n)$ such that they fulfill the usual Poisson brackets: \...
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### What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?

One of the most important results of Classical Mechanics is Liouville's theorem, which tells us that the flow in phase space is like an incompressible fluid. However, in the phase space formulation ...
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### What is the difference between configuration space and phase space?

What is the difference between configuration space and phase space? In particular, I notices that Lagrangians are defined over configuration space and Hamiltonians over phase space. Liouville's ...
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### Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
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### Is there any uncertainty between mass and proper length or time?

I was trying to naively draw a parallel between special relativity and the Heisenberg uncertainty principle. I try to understand uncertainty principle as a consequence of 4-position and 4-momentum ...
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### Does Hamilton Mechanics give a general phase-space conserving flux?

Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltonian theory like the flux of an ideal fluid, which doesn't change ...
Given a 1 d.o.f Hamiltonian $H(q,p)$ what is the general procedure for finding action angle variables $(I, \theta)$? I have read the Wikipedia page on action angle variables and canonical transforms ...