All Questions
8 questions
2
votes
0
answers
53
views
Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials
I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses:
Coulomb potential with a ...
- 21
11
votes
0
answers
233
views
Mathematical pendulum and $\mathbb C P^n$
I am very puzzled by the following remark on p.346 in Arnold's book "Mathematical methods of classical mechanics":
Another method of construction the same symplectic structure on complex ...
- 5,063
2
votes
0
answers
195
views
How to check conditions for Liouville-Arnold theorem? [closed]
Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem:
Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...
- 679
1
vote
1
answer
330
views
Global reduction of Hamiltonian with an integral of motion (Poincare' reduction)
This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question.
Background
I refer to the following concepts:
Liouville ...
- 1,301
7
votes
2
answers
2k
views
Practical example of Hamiltonian reduction
I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p_1, \...
- 31.5k
2
votes
0
answers
491
views
How to make sense of the Euler Lagrange equations for an infinite action?
The Euler–Lagrange equation is an equation satisfied by a function $q$, which is a stationary point of the functional
$S(\boldsymbol q) = \int_a^b L(t,q(t),\dot{q}(t))\, \mathrm{d}t$
Say we have an ...
- 366
7
votes
2
answers
2k
views
Momentum a cotangent vector
Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.
Furthermore, many sources ...
- 231
4
votes
1
answer
396
views
Weinstein's local classification of Lagrangian foliations
In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I ...
- 576