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9 questions
1
vote
0
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108
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Stability of rigid bodies spinning around $z$-axis under gravity
Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
- 807
0
votes
0
answers
90
views
Numerical detection of Cantori
It is known that as parameters vary in Hamiltonian system, KAM tori can break [1,2].
How to construct numerically the breaking tori?
The most relevant paper that I could find is [3,4].
But it uses ...
- 24.2k
6
votes
0
answers
469
views
An algebraic Hamiltonian vector field with a finite number of periodic orbits (2)
Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
- 7,583
4
votes
1
answer
235
views
Dynamical analogue of Morse theory
Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property:
For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X_H$ on $H^{...
- 7,583
1
vote
0
answers
77
views
What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?
When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
- 183
2
votes
0
answers
129
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Is the interpolating Hamiltonian flow of an exact near-identity symplectic map globally defined?
It is well-known that an analytic near-identity map $\bar{x} = F_{\epsilon}(x) = x + \epsilon f(x) + O(\epsilon^{2})$ may be embedded into the flow of a differential equation, and if that map is ...
- 993
2
votes
0
answers
191
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Geometric properties of solutions of Hamiltonian system
Context : We are interested in the following dynamic with state $(q,\varphi)$
$$
\dot q = \varepsilon F(q,\varphi), \quad \dot \varphi = \omega(q) + \varepsilon G(q,\varphi)
$$
($\varepsilon >0$ ...
- 141
9
votes
2
answers
648
views
An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)
Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question.
Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ(...
CommunityBot
- 1
1
vote
0
answers
65
views
Id monodromy in hamiltonian dynamics
In my problem I have non autonomous Hamiltonian which depends on 2 parameters (pretty close to oscillator Hamiltonian, $(a+b\cos t +1) p^2+(a+b\cos t-1)q^2$, $a,b$ - parameters). From numerical ...
- 11