All Questions
11
questions
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Hamiltonian flows and Poisson Brackets confusion
I have been using results from this paper in calculations. In sections 2.4 and 3.4 they perform a canonical transformation into new coordinates consisting of constants of motion.
My question is ...
- 298
9
votes
4
answers
2k
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Time evolution operator in classical mechanics?
Hamilton's equation can be written in terms of Poisson brackets, as follows:
$$\dot{q} = \{q,H\}$$
$$\dot{p} = \{p,H\}$$
where $H$ is the Hamiltonian of the system. Now, wikipedia says that the ...
- 1,131
1
vote
1
answer
318
views
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
answers
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
0
votes
1
answer
214
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Assignment of energy functions to flows is "equivariant"?
I am trying to understand the 2012 blog post What is a symplectic manifold, really?
It says (with correction of a typo in the second point):
If $f: M \to \mathbb{R}$ is a smooth compactly ...
- 103
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
6
votes
1
answer
5k
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Hamiltonian flow?
I was wondering what the Hamiltonian flow actually is?
Here is my idea, I just wanted to know if I am correct about this.
So let $(x(t),p(t))' = X_{H}(x(t),p(t))$ are the Hamilton's equations and $...
- 97
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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