All Questions
Tagged with hamiltonian differential-geometry
10
questions
0
votes
0
answers
36
views
When is the derivative of Hamilton flow respect to initial conditions independent of time?
Consider a Hamiltonian system with coordinates $\Gamma^A=(q^i,p_i)$ and let $X^A(s,\Gamma_0)$ be the Hamiltonian flow (i.e. a solution to Hamilton's equations) parametrized by time $s$ and initial ...
- 4,827
0
votes
1
answer
92
views
Why inverse flow of separable Hamiltonian with even kinetic energy can be written like this?
Why is it true that the inverse of the flow of a separable Hamiltonian with even kinetic energy can be written as $\phi_N \circ \varphi_t \circ \phi_N$ where $\varphi_t$ is the flow of the Hamiltonian ...
- 247
0
votes
1
answer
207
views
Definition of time-reversibility of flow of Hamilton's equations
I cannot find a good, simple definition of time-reversibility of the flow $\phi_t$ of Hamilton's equations
$$
\dot{z} = J^{-1}\nabla_z H(z) \quad \text{where} \quad z = (q, p)^\top \quad \text{and} \...
- 247
1
vote
1
answer
56
views
Is an integrator with unit Jacobian determinant sympletic?
Consider a numerical integrator $\phi(q, p)$ having $|\text{det } \phi'| = 1$. Can we say the integrator is sympletic, that is that
$$
\phi'^\top J^{-1} \phi' = J^{-1}
$$
where
$$
J^{-1} = \begin{...
- 247
2
votes
2
answers
725
views
Quantum particle moving on the surface of cylinder
I have a problem with a spinless particle moving on the lateral surface of a cylinder of radius $r$.
If no Hamiltonian is given, is $H=\frac{p^2}{2m}$ only?
What are the Hamiltonian's eigenfunctions ...
- 941
2
votes
1
answer
327
views
Rewriting the Laplacian on a curved manifold
I guess there is a sense in which the following is true:
"The Laplacian written on a Riemannian manifold $(M,g)$ can be seen as adding a coordinate dependent mass field to the Laplacian on ...
- 131
5
votes
1
answer
269
views
The WKB approximation and the Cotangent bundle
When we say (see pag. 9 of Lectures on the Geometry of Quantization) that
the image of the differential of the phase function lies in the level set of the classical Hamiltonian
is it simply ...
- 139
0
votes
1
answer
55
views
Hamiltonian as differential manifold
I know that phase space $(q^i, p^i)$ can be treated as manifold. But for me defining hamiltonian as a function also leads to new manifold $(q^i, p^i, H(q^i,p^i))$. Like map from open set $(q^i, p^i) \...
- 21
1
vote
0
answers
1k
views
Hamiltonian for particle moving in a sphere
Context: I know that if we have a particle (say, with unit mass) moving in the plane $\Bbb R^2$ subject to a spherically symmetric potential $V\colon \Bbb R^2 \to \Bbb R$, it will move along the ...
- 545
0
votes
0
answers
208
views
General geometric interpretation for the Hamiltonian and of the cases when it is not the total energy of the system
What is the geometric interpretation for the Hamiltonian? Also, is there geometric interpretation of when and why it is not equal to the total energy of the system?
Lastly, what is the most general ...
- 7,720