# Questions tagged [hamiltonian-mechanics]

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93
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### 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}...

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### 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 ...

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### Integrability of Schroedinger's equation

Consider the periodic nonlinear Schrödinger equation
$$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$
where $\mathbb{T}:= \mathbb{R}/\...

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### Poisson bracket on $T^*T\mathrm{SU}(1,1)$

Consider the cotangent bundle of the tangent bundle $T^*TG$ of a Lie group $G$. Denote its the Lie algebra by $\mathfrak{g}$. By left translations, we have the trivialization $T^*G \cong G \times \...

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### 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^{...

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### 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 ...

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### Why is this Hamiltonian flow of the Vlasov equation well defined?

Lions and Paul claim in their 1993 paper "Sur les mesures de Wigner" that the Hamiltonian flow
$$\dot{x} = \xi, \quad \dot{\xi} = - \nabla V(x) $$
of the Vlasov equation
$$\partial_t f + \xi ...

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### Integral expression for the Poisson bracket

I already asked this in the physics forum but without much attention, so I thought it might attract more attention here.
Is there an integral expression for the Poisson bracket that can be derived ...

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### 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 ...

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### Hamiltonian particle system and its frequency domain

I am interested in the following question.
So let suppose we have finite number of point particles on plane $\mathbb{R}^2$.
We can assume that every $j$ point is represented by Dirac delta function $\...

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### Gradient descent relaxation dynamics of a Euler-Lagrange equation

I want to minimize the functional
$$
F=\int{L(u)}dx,
$$
where $L= u_x^2-u^2$ is the Lagrangian function of the functional. Even if its Euler-Lagrange equation is easily found and solved, I want to try ...

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### What is the motivation of contact Hamiltonian equation

I've just checked that this is constructed to mimic the ordinary Hamiltonian equation in symplectic geometry. There are several literatures, and they use
$$
\eta(X_H) = -H\\
\mathrm{d}\eta(X_H,-) = \...

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### Uniform continuity of Hamiltonian flow

Let $h \in C^2_{\mathrm{ub}}(\mathbb{R}^{2n})$, where $C_{\mathrm{ub}}^k$ consists of $C^k$-functions that are bounded and uniformly continuous along with their derivatives up to $k$th-order.
It is ...

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### In the context of field-theoretic classical Lagrangian mechanics, can we choose the Lagrange multipliers to be time-independent? - from Physics SE

I originally posted this question on Physics SE, but I think it is more like a math question since I need rigorous justification.
Could anyone please provide any insight to the below question:
Let us ...

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### 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 ...

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### From time-dependent Hamiltonians to time-dependent symplectic/Poisson structures

Let $(M,\{.,.\})$ be a smooth Poisson manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.
Question: Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and smooth parameter-dependent Poisson ...

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### Hamiltonian, energy, and conservation laws of nonlinear PDEs

In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...

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### Lie groupoids as symmetries of mechanical systems?

Lie groups are well studied as symmetries of mechanical systems in symplectic/Poisson geometry. For instance, if $G$ acts freely and properly on a mechanical system modeled by a symplectic manifold $(...

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### 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: ...

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### algebraic momentum map

Let $T$ be a linear algebraic torus over $\mathbb C$ and $X$ be a smooth quasi-projective symplectic $T$-variety. Also, assume that the action of $T $ is free and $X/T$ exists as a smooth variety. Is ...

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### Mechanical systems with their configuration space being a Lie group

Cross-posted from Physics.SE
In Marsden, Ratiu - Introduction To Mechanics And Symmetry there is a certain focus on reducing cotangent bundles of Lie groups. More precisely, if $G$ is a Lie group, ...

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### 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 ...

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### Hamilton equations for Classical Field Theory

This is a second part of my previous question. I'm trying to figure it out by myself how to deduce Hamilton's equations in classical field theory as it is usually obtained in physics books.
Notation: ...

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### Applications of Hamiltonian formalism to classical mechanics

In many courses in theoretical classical mechanics Hamiltonian formalism takes an important place. However I did not see it applied to problems of classical mechanics (unless one expands the scope of ...

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### 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 \...

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### 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 ...

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### Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...

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### 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 ...

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### Existence results for Lagrangian solutions to the Incompressible Euler Equation?

It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with $X \in H^1_t$ (i.e. has weak time ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...