# Questions tagged [hamiltonian-mechanics]

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66
questions

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80 views

### Is there a notion of symplectic maps between spaces of volume forms on phase spaces?

For a $n$ dimensional smooth manifold $M$, I consider the cotangent bundle $T^*M$ with the canonical symplectic form $\omega$.
A symplectic map $\phi : T^*M \to T^* M$ is a map which leaves the ...

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53 views

### Quantization of a non Hamiltonian flow

If one considers a classical Hamiltonian flow $\partial_t \boldsymbol{x} = \partial_p H(\boldsymbol{x},\boldsymbol{p})$, $\partial_t \boldsymbol{p} = -\partial_x H(\boldsymbol{x},\boldsymbol{p})$, the ...

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**1**answer

122 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 ...

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49 views

### Mathematical construction: ADM formulation in general relativity

I'm doing my undergraduate thesis and now I'm looking for references that presents ADM Formulation in general relativity mathematically.
I studied the basics of general relativity theory by O'Neill ...

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274 views

### A question in elementary differential geometry

Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation
$$ \partial_t u + |\nabla u|^2 = 0. $$
Can we give a precise estimate of a ...

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81 views

### Geometrical proof of Noether Theorem [duplicate]

I am reading a very nice Physics book "The standard model in a nutshell" by D.Goldberg and just read there a mention to Noether Theorem. Of course I knew this outstanding theorem very well from ...

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**1**answer

43 views

### Invertibility of the characteristic flow in Hamilton-Jacobi equations

We are in the context of Hamilton Jacobi equations, in particular I was reading the characteristic method. We want to solve the problem of the special form (Hamiltonian only depending on the "$p$" ...

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53 views

### Infinitesimal orbit type decomposition of Hamiltonian $G$-manifolds

Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. ...

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189 views

### Asymptotic of an integral

Let
\begin{equation*}
V(x) = -\big(2-\sin(2\pi x) - \sin(2\pi \sqrt{2}x)\big)^\gamma
\end{equation*}
for some $\gamma \in (0,1]$. Define for each $r<0$ the number
$$a_r = \min\{a>0: V(a) = ...

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104 views

### Why does the Lax pair formalism look so similar to the Hamiltonian equations, and what is the significance of this?

If we have a Lax pair for a system, which we'll call operators $L$ and $B$, then the system
\begin{align*}L\psi&=\lambda\psi\\
\psi_t&=B\psi\end{align*}
has as its integrability condition ...

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7k views

### Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?

Given a spherically symmetric potential $V: {\bf R}^d \to {\bf R}$, smooth away from the origin, one can consider the Newtonian equations of motion
$$ \frac{d^2}{dt^2} x = - (\nabla V)(x)$$
for a ...

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135 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 $2n$-...

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58 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 ...

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**1**answer

223 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 ...

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**2**answers

479 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, \...

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115 views

### Is this integral zero?

I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation.
Consider $\mathfrak{g}=C^{\infty}(M)$ and the dual $\mathfrak{g}^*=\...

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95 views

### What is the relation between the different generating functions thought as finite approximations of action functionals

In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on $\mathbb{R}^{2n}$. The idea is that a Hamiltonian isotopy can be ...

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321 views

### Constants of motion for Droop equation

There is an important ODE system in biochemistry, Droop's equations:
$$s'=1-s-\frac{sx}{a_1+s}$$
$$x'=a_2\big(1-\frac{1}{q}\big)x-x$$
$$q'=\frac{a_3s}{a_1+s}-a_2(q-1)$$
Relatively easy one finds a ...

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votes

**1**answer

229 views

### Non-Hamiltonian actions in physics

I was reading the following article when I came across the interesting sentence
"non-Hamiltonian [symplectic group] actions also occur in physics"
I took a cursory look at the article cited but ...

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226 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 ...

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73 views

### Effective actions by non-commutative groups have non-commuting fundamental vector fields?

I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :)
Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ ...

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80 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 ...

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**1**answer

209 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 ...

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334 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.

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203 views

### Proving that system is Hamiltonian

This question is moved from math stackexchange, seems like it is a more advanced question. Here the link from the original question: https://math.stackexchange.com/questions/2666194/proving-that-...

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**1**answer

280 views

### Symplectic reduction of 4-manifolds with circle actions

Let $(M,\omega)$ be a $4$-dimensional closed symplectic manifold. Assume there exists a Hamiltonian $S^1$-action on $M$, let $\mu:M \to \mathbb{R}^*$ be its moment map and let $M_{\text{red}}=\mu^{-1}(...

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211 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 ...

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25 views

### First return map in complex 2DOF Hamiltonian systems

The standard way to construct the first return map around a periodic orbit in real 2DOF Hamiltonian systems is the following:
We choose a periodic orbit and a point on it.
We restrict the system on ...

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349 views

### When is a Divergence-Free Vector Field on the Tangent Bundle of a Riemannian Manifold Hamiltonian?

(Reposted from https://math.stackexchange.com/questions/2589600/when-is-a-divergence-free-vector-field-on-the-tangent-bundle-of-a-riemannian-man) Starting with a closed, connected Riemannian manifold $...

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156 views

### Isolated periodic trajectories of Hamiltonian systems

Is there any example of an autonomous Hamiltonian system with a periodic trajectory isolated in the whole phase space? The Poincar\'e map of such a trajectory within its energy level should be very ...

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**1**answer

523 views

### What are the compact Lagrangian submanifolds of a twisted cotangent bundle?

In Hamiltonian dynamics and symplectic geometry a twisted cotangent bundle is the cotangent space $T^*N$ of a closed (compact without boundary) $n$-manifold $N$ equipped with a twisted symplectic ...

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**1**answer

145 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 ...

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144 views

### Most probable path for stochastic Hamiltonian systems

It is known that for a real valued stochastic process $X_t$ satisfying
$$
d X_t = b(X_t) d t + \sigma d W_t
$$
where $W$ is real valued Wiener process, the equation for the most probable path from ...

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**1**answer

119 views

### Infinitesimal generators and conserved quantities (Schrodinger type evolution)

First, I'm no expert in symmetry analysis of evolution equations and so I apologize if this post is a bit of a cobble. The question I have is about the evolution of $\psi: \mathbb{R}^{1+1}\to \mathbb{...

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**1**answer

368 views

### Sampling point from the surface of an n-dimensional ellipsoid with uniform distribution

I am wondering if exist an efficient computational method for sampling points belonging to the surface of an ellipsoid in $n$-dimensional space with n even, I am thinking in the phase space of a ...

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**4**answers

11k 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,...

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**1**answer

519 views

### Why is every Hamiltonian system locally integrable?

It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional ...

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112 views

### Reference request: Liouville integrability of a torus action of small dimension on a symplectic manifold

Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by $k$, may be less than $n$. The generators of the action will ...

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**1**answer

461 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,......

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**0**answers

75 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 ...

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306 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 ...

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**1**answer

301 views

### Generalizing HJB equation for a terminal stopping time

The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation:
Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[...

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**0**answers

180 views

### Geometric properties of solutions of Hamiltonian system

Context : We are interested in the following dynamic with state $(q,\varphi)$
$$
\dot q = \varepsilon F(q,\varphi), \quad \dot \varphi = \omega(q) + \varepsilon G(q,\varphi)
$$
($\varepsilon >0$ ...

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210 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 $...

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59 views

### Comparison inequalities for Hamiltonian mechanics with convex potential - analogue to Rauch's theorem?

I'm asking about an area (Hamiltonian mechanics) that I don't know at all well; thus, I keep the question somewhat vague.
In differential geometry, there are a number of results saying that geodesics ...

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**1**answer

464 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 ...

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33 views

### The isotropy group for the Euler-Lagrange vector-fields

Let $Q$ be a manifold, and let $X_{EL}$ be a second order vector-field on $TQ$ derived from the Euler-Lagrange equation,
$$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q} } \right) - \frac{ \...

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250 views

### Deformation quantization of Poisson bracket without star-product

Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...

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72 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}...

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**1**answer

131 views

### Points with finite stabilizer in Hamiltonian torus actions

Atiyah-Guillemin-Sternberg theorem asserts that the image of the moment map $\mu$ for a Hamiltonian $(S^1)^m$-action on a smooth compact symplectic manifold $(M^{2n},\omega)$ is a convex polytope of $\...