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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 ...
Zhang Yuhan's user avatar
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 ...
0x11111's user avatar
  • 593
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^{...
Ali Taghavi's user avatar
1 vote
0 answers
76 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 ...
user135626's user avatar
2 votes
0 answers
129 views

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 ...
Vitaly's user avatar
  • 21
2 votes
0 answers
191 views

Geometric properties of solutions of Hamiltonian system

Context : We are interested in the following dynamic with state $(q,\varphi)$ $$ \dot q = \varepsilon F(q,\varphi), \quad \dot \varphi = \omega(q) + \varepsilon G(q,\varphi) $$ ($\varepsilon >0$ ...
Smilia's user avatar
  • 141
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 ...
Ali Taghavi's user avatar
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 ...
cptpwnage's user avatar
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(...
Ali Taghavi's user avatar