All Questions
8
questions
1
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1
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318
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On the Hamiltonian vector fields of classical Hamiltonian mechanics
Notation: I denote phase space as the symplectic manifold $(M,\omega)$, in which $\omega=\sum_i\mathrm dp_i\wedge\mathrm dq_i$ in canonical coordinates.
In definitions of Hamiltonian vector fields I ...
- 6,963
6
votes
3
answers
2k
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Paths in phase space can never intersect, but why can't they merge?
Page 272 of No-Nonsense Classical Mechanics sketches why paths in phase space can never intersect:
Problem: It seems to me this reasoning only implies that paths can never "strictly" ...
- 337
1
vote
2
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333
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Analogues to Hamilton's equations in Infinitesimal Canonical Transformations
This is from chapter 4 of David Tong's notes on Classical Dynamics (Hamiltonian Formalism).
Let's say you make an infinitesimal canonical transformation (with $\alpha$ as the infinitesimal parameter) ...
- 773
1
vote
0
answers
65
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Quantum mechanics in phase space - what are coordinate components?
I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
- 123
5
votes
1
answer
756
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Liouville's volume theorem in differential forms language
I will cast my question mostly in words. I usually find Liouville's volume theorem cast in two forms:
Easy way (without differential forms language): Phase space volume remains preserved under ...
- 837
8
votes
1
answer
702
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Can any symplectomorphism (1 Definition of canonical transformation) be represented by the flow of a vectorfield?
For this question I will use the definition that a canonical transformation is a map $T(q,p)$ from the phase space onto itself, which leaves the symplectic 2-form invariant (which is the definition of ...
- 6,763
3
votes
1
answer
71
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Two first integrals of an hamiltonian field $X_{H}$ are independent $\det \left[ \frac{\partial F_{i}}{\partial p_{k}} \right] \neq 0$
I want to understand how it is established if two first integrals of an hamiltonian field $X_{H}$ are independent.
One hypothesis is:
Considering two first integrals $F(q^i,p_k)$
$$\det \left[ \...
- 585
11
votes
2
answers
1k
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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 ...
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