All Questions
Tagged with hamiltonian-mechanics ds.dynamical-systems
30 questions
27
votes
4
answers
13k
views
Hamiltonian, Lagrangian and Newton formalism of mechanics
If my thinking is wrong please let me know. I have little knowledge on beyond-college physics.
For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,...
- 8,071
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(...
- 366
9
votes
1
answer
728
views
When does a Lagrangian dynamical system have an equivalent Hamiltonian description?
Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$
(i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q_{i},\dot{q}_{i},t)$, $i=1,2,......
- 7,746
7
votes
1
answer
719
views
Integrable systems and Arnol'd - Liouville theorem
A system with a $2n$-dimensional phase space is Liouville-integrable if it admits $n$ independent first intgrals in involution.
Here integrable means that you can, in some way, solve the equations of ...
- 71
7
votes
1
answer
267
views
Uniqueness of Birkhoff Normal Form and KAM theory for Symplectomorphims
I am starting to work with Hamiltonian Dynamics and I have been taking a look at some of the basic stuff in KAM theory. I have posted part of this question at MSE but as I did not get any response I ...
6
votes
2
answers
3k
views
Poincaré recurrence and its implications for statistical physics and the arrow of time
A very important theorem in mathematical physics is Poincaré’s recurrence theorem.
As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for ...
- 1,524
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 ...
- 366
4
votes
3
answers
668
views
Reference Request: KAM Theory
I intend to learn KAM Theory. Could you please suggest me a good book on KAM Theory to begin with, where main results are discussed with complete proofs.
Thank you.
4
votes
1
answer
235
views
Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus
Again a very simple question. I currently hold two contradictory ideas in my head
1) A hamiltonian diffeomorphism of a torus necessarily has fixed points
2) most hamiltonian actions on a torus in an ...
- 989
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^{...
- 366
4
votes
1
answer
326
views
Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$
Define the map $$P:TS^{n}\to S^{n} \;\;\;\text{by}\;\; P((x,v))=\frac{x+v}{\parallel x+v \parallel}$$ where $$TS^{n}=\{(x,v)\in S^{n} \
\times \mathbb{R}^{n+1}\mid v \perp x \}$$
This map is ...
- 366
4
votes
1
answer
299
views
Symplectic forms and sign of eigenvalues
This question has come out while reading J. Moser "New Aspects in the Theory of Stability
of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of ...
- 255
4
votes
0
answers
104
views
Why can every twist map be realized as the time-1 map of a time-dependent Hamiltonian?
if have problems getting my head around the following claim made
by Moser in "Monotone twist mappings and the calculus of variations" and Gole in "Symplectic twist maps".
Setting:
Let $F : \mathbb{R}...
- 41
3
votes
1
answer
197
views
Exact solution to a periodic linear ODE sought
We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these ...
- 33
3
votes
1
answer
679
views
Perturbed vs. unperturbed Hamiltonian system
Let's take a time-periodic Hamiltonian $H(t,x,y)$ on $\mathbb{R}^2$ and
apply an arbitrarily small time-independent perturbation to $H$ via
$$
\tilde H (t,x,y) = H(t,x,y) + \epsilon V(x,y),
$$
where $...
- 81
3
votes
0
answers
143
views
Is composition of discrete Hamiltonian flows integrable?
Consider $\Bbb{R}^2$ with the usual symplectic form $$\omega = dx \wedge dy$$
For a function $H \colon \Bbb{R}^2 \to \Bbb{R}$, let $X_H$ be the Hamiltonian vector field. Then the map $\Bbb{R}^2 \to \...
- 213
3
votes
0
answers
245
views
Periodic orbit for certain Hamiltonian on the tangent bundle
In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point.
Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...
- 366
2
votes
1
answer
102
views
On the existence of regular orbit cylinders
Let $(M,\omega,H)$ be a Hamiltonian system and assume that $\gamma$ is a periodic orbit on a regular energy hypersurface. Then the regular orbit cylinder theorem (see for example Abraham/Marsden: ...
- 548
2
votes
0
answers
101
views
Persistence of KAM tori as a function of dimension
I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here.
In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...
- 121
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 ...
- 21
2
votes
0
answers
191
views
Geometric properties of solutions of Hamiltonian system
Context : We are interested in the following dynamic with state $(q,\varphi)$
$$
\dot q = \varepsilon F(q,\varphi), \quad \dot \varphi = \omega(q) + \varepsilon G(q,\varphi)
$$
($\varepsilon >0$ ...
- 141
2
votes
0
answers
143
views
Is it true that a nondegenerate minimizing periodic orbit of mechanical Hamiltonian system is hyperbolic
Consider mechanical Hamiltonian system of the form
$$H(p,q)=\dfrac{\Vert p\Vert^2}{2}+V(q),\quad (q,p)\in T^*\mathbb T^n.$$
Here we suppose the periodic orbit $\gamma$ minimizes the Lagrangian ...
- 197
1
vote
0
answers
108
views
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
1
vote
0
answers
41
views
The boundedness of dynamical systems discretized from Hamiltonian systems
Let $H(p,q) = T(q) + U(p)$ be a Hamiltonian function that defines a Hamiltonian system, i.e.,
\begin{align}
&\frac{dp}{dt} = \frac{\partial H}{\partial q}(p,q) = \frac{dT}{dq},\\
&\frac{dq}{dt}...
- 47
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
1
vote
0
answers
67
views
Reduced master equation for a multistable Hamiltonian dynamical system
I am looking for rigorous results on the derivation of a reduced master equation for a (possibly stochastic) Hamiltonian dynamical system with a coercive potential energy term with multiple local ...
- 3,873
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
0
votes
1
answer
214
views
Hamilton equations-Symplectic scheme [closed]
We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...
- 111
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 ...
- 593
0
votes
0
answers
94
views
On the measure of regular and chaotic regions in a phase space
Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...
- 255