Questions tagged [hamiltonian-mechanics]
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103 questions
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Substitution of the old variable to a generating function
Assume that we have one of the 4 types of generating functions (say F(p, Q)) which defines a symplectomorphism $T$:
$(P,Q) = (u(p,q), v(p,q))$. Now we can make a substitution and get a function $G(p,q)...
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77
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Projecting phase space Liouville tori to configuration space in integrable systems
My question concerns whether or not the topology of a trajectory in the configuration space of an integrable system, or rather the area of configuration space accessible by trajectories can be ...
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137
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When the integrable system defines a moment map?
Assume $\mathcal{M}$ is a compact symplectic $2n$-dimensional manifold with a Hamiltonian action of the torus $\mathbb{T}^n$. Given a family of functions $F=(f_1,\ldots,f_n)$ defining an integrable ...
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44
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When lagrangian fibrations are equivalent?
Given a $2n$-dimensional symplectic manifold $\mathcal{M}$ and two different lagrangian fibrations $\pi_1:\mathcal{M}\rightarrow \Gamma_1$ and $\pi_2:\mathcal{M}\rightarrow \Gamma_2$, with $\Gamma_1, \...
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79
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Liouville-Arnold and fibration relative to a convex polytope
Liouville-Arnold's theorem indicates that given a Hamiltonian torus action on a manifold and a set of $n$ functions $F$ from the manifold to $\mathbb{R}^n$ defining an integrable system, the pre image ...
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2
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323
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Generating function for composition of symplectomorphisms
Given two symplectomorphisms $T_1: (q_1, p_1):\longrightarrow (q_2, p_2)$ and
$T_2: (q_2, p_2):\longrightarrow (q_3, p_3)$ and corresponding generating functions $F_1(q_1, q_2)$ and $F_2(q_2, q_3)$, ...
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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
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153
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Spectrum of an almost Hamiltonian matrix
I have a complex-valued block matrix $N=\begin{bmatrix}
A & B \\
C & -A^*
\end{bmatrix}$, where $A$ is diagonal, $B=B^*$, and $C$ is rank-1 but not Hermitian.
If $C$ were Hermitian, $N$ would ...
- 116
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52
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How to prove a solution to Hamilton's equations on the tangent bundle is second order?
Let $Q$ be a manifold Let $q^\alpha$ for $\alpha \in 1 .. n$ be a chart of $Q$.
The tangent bundle $TQ$ has a chart $q^\alpha, \dot{q}^\alpha$. It has an almost-tangent structure:
$$
J = dq^\alpha ...
- 238
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264
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How to express the Euler-Lagrange equation in arbitrary coordinates where $\omega \neq \sum dp_i \wedge dq_i$
I posted my questions in a previous post MO, but it seems that a more refined version for question on the Euler-Lagrange equation is needed. So, I post my question again.
In standard symplectic ...
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How does the symplectic form $\omega$ manifests itself in the Euler-Lagrange equation? + Extreme confusion with time
Let $\omega$ be a symplectic manifold on $\mathbb{R}^n$ and the smooth function $H : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a Hamiltonian. For $p,q \in \mathbb{R}^n$ let us assume that
\...
- 3,487
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The boundedness of dynamical systems discretized from Hamiltonian systems
Let $H(p,q) = T(q) + U(p)$ be a Hamiltonian function that defines a Hamiltonian system, i.e.,
\begin{align}
&\frac{dp}{dt} = \frac{\partial H}{\partial q}(p,q) = \frac{dT}{dq},\\
&\frac{dq}{dt}...
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90
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Numerical detection of Cantori
It is known that as parameters vary in Hamiltonian system, KAM tori can break [1,2].
How to construct numerically the breaking tori?
The most relevant paper that I could find is [3,4].
But it uses ...
- 593
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429
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Integrability of Schroedinger's equation
Consider the periodic nonlinear Schrödinger equation
$$-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,$$
where $\mathbb{T}:= \mathbb{R}/\...
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Poisson bracket on $T^*T\mathrm{SU}(1,1)$
Consider the cotangent bundle of the tangent bundle $T^*TG$ of a Lie group $G$. Denote its the Lie algebra by $\mathfrak{g}$. By left translations, we have the trivialization $T^*G \cong G \times \...
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235
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Dynamical analogue of Morse theory
Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property:
For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X_H$ on $H^{...
- 366
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101
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Persistence of KAM tori as a function of dimension
I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here.
In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...
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90
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Why is this Hamiltonian flow of the Vlasov equation well defined?
Lions and Paul claim in their 1993 paper "Sur les mesures de Wigner" that the Hamiltonian flow
$$\dot{x} = \xi, \quad \dot{\xi} = - \nabla V(x) $$
of the Vlasov equation
$$\partial_t f + \xi ...
- 469
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157
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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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53
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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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308
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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 ...
- 159
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2
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218
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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,-) = \...
- 285
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207
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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 ...
- 1,514
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351
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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 ...
- 1,959
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369
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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 ...
- 159
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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$$
For a function $H \colon \Bbb{R}^2 \to \Bbb{R}$, let $X_H$ be the Hamiltonian vector field. Then the map $\Bbb{R}^2 \to \...
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Mathematical pendulum and $\mathbb C P^n$
I am very puzzled by the following remark on p.346 in Arnold's book "Mathematical methods of classical mechanics":
Another method of construction the same symplectic structure on complex ...
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536
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Hamiltonian dynamics on cotangent bundle
I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
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What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?
When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
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Existence results for Lagrangian solutions to the Incompressible Euler Equation?
It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with $X \in H^1_t$ (i.e. has weak time ...
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Is there a notion of symplectic maps between spaces of volume forms on phase spaces?
For a $n$ dimensional smooth manifold $M$, I consider the cotangent bundle $T^*M$ with the canonical symplectic form $\omega$.
A symplectic map $\phi : T^*M \to T^* M$ is a map which leaves the ...
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1
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Exact solution to a periodic linear ODE sought
We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these ...
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A question in elementary differential geometry
Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation
$$ \partial_t u + |\nabla u|^2 = 0. $$
Can we give a precise estimate of a ...
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136
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Geometrical proof of Noether Theorem [duplicate]
I am reading a very nice Physics book "The standard model in a nutshell" by D.Goldberg and just read there a mention to Noether Theorem. Of course I knew this outstanding theorem very well from ...
- 4,361
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Invertibility of the characteristic flow in Hamilton-Jacobi equations
We are in the context of Hamilton Jacobi equations, in particular I was reading the characteristic method. We want to solve the problem of the special form (Hamiltonian only depending on the "$p$" ...
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Infinitesimal orbit type decomposition of Hamiltonian $G$-manifolds
Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. ...
- 1,942
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Asymptotic of an integral
Let
\begin{equation*}
V(x) = -\big(2-\sin(2\pi x) - \sin(2\pi \sqrt{2}x)\big)^\gamma
\end{equation*}
for some $\gamma \in (0,1]$. Define for each $r<0$ the number
$$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 ...
- 114k
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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 ...
- 3,873