Questions tagged [hamilton-equations]

Use this tag for questions related to Hamilton's equations.

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Application of the Arnold-Liouville theorem

We are given a system with Hamiltonian $\displaystyle H(\phi,r,p_{\phi},p_r)=\frac{p_{\phi}^2}{2r^2}+\frac{p_r^2}{2}-\frac{1}{r}$. Since $\displaystyle \frac{\partial H}{\partial \phi}=0$, it ...
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24 views

derivative of a sum - derivation of Boltzman equation

Let $x_i$ is a position vector (for simplicity in 1D) of an $i$-th particle. $V(x_i,x_j)=\phi(|x_i - x_j|)$ is some function that depdends only on the distance between the two particles. I would like ...
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45 views

Second derivatives, Hamilton and tangent bundle of tangent bundle TTM

I'm learning the Hamilton formalism of classical mechanics, where a second order differential equation is formalized as two first order differential equations on the cotangent bundle of the ...
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Hamilton-Jacobi equation and Action Functional

Let the action functional $S[q]$ given by \begin{equation}\label{eq16} S[q]=\int\limits^{t_2}_{t_1}L\left(q^i(t),\dot{q}^i(t)\right)dt \end{equation} Also, we know that using Legendre Transform ...
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22 views

Dynamic optimisation / optimal control where control is a function of state

I am familiar with the usual optimal control problem of the form: $$ \min_{u(t)}\int_{t_0}^{t_1}{f(t,x(t),u(t))}dt\\ \text{s.t.}~\dot{x}(t)=g(t,x(t),u(t))\\ \text{given } x(0), t_0, t_1. $$ I am ...
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29 views

Minimum and Maximum on first Integral

Given the Hamiltonian $H(x,y)=\frac{1}{2}y^2+\frac{1}{4}x^4$ of a Hamiltonian vector field $X(x,y)$, where $\dot{H}= DH(x,y)X(x,y)=y^2(1-x^2-y^2).$ Find $0<h_1<h_2$ such that, for all $0<h<...
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1answer
33 views

What does it mean for a function to be constant along solutions?

I've come across a question that asks to prove that a function is constant along solutions. The only other bit of information is that the function is Hamiltonian. My question is what does it mean ...
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47 views

Proving Hamiltonian symplectomorphism commutes with given symplectomorphism

I'm trying to prove this identity which is mentioned at the very beginning of this paper by Dostoglou and Salamon "Self-Dual instants and holomorphic curves". Let $(M,\omega)$ be a closed symplectic ...
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50 views

1-Form and 2-Form in cotangent bundle with time dependecy

I have these question but first of all a bit of context. I have a Lagrangian $L(q,\dot q,t)\in C^\infty(\textbf{T}M \times \mathbb{R})$. We know that the taulogical 1-form in $\textbf{T}M$ is $\theta=...
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41 views

Passage to limit in differential equation

we consider the equation \begin{equation} \partial_t u (t , x) + v (t, x) \cdot \nabla u ( t, x) = \kappa \Delta u (t , x) + F ( t, x, u (t , x) ) \qquad \mbox{in} \ \ \mathbb{R}_+ \times \mathbb{R}^...
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How do I know when using the Euler Lagrange equations that I have found a minimum of the action

When looking at a physical system where I have the lagrangian. How do I know I have found a minimum using the EL equations (since according to the principle of least action we need a minimum) as ...
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83 views

Example of a symplectic but non-Hamiltonian vector field on $\mathbb{T}^{2n}$

I want to show that there exists a symplectic vector field on the $2n$ torus $\mathbb{T}^{2n}$, endowed with the unique symplectic form $\omega$ that pullsback to the canonical symplectic form $\...
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37 views

canonical transformation

A system of Hamiltonian equations in generalized coordinate $q$ and momentum $p$ is given by $$ \frac{d q}{d t}=\frac{\partial H}{\partial p}, \ \ \ \ \frac{d p}{d t}=-\frac{\partial H}{\partial q} $$...
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27 views

Can you think of a good way to solve this Hamiltonian function with two control variables and one state

I have to following functions describing an economic macro model with polution: $ \begin{align} Y_t &=K_t^\alpha(A_tL_t)^{1-\alpha}z_t, \; \; L_t=1, \; \; z_t \in [0,1] \\ P_t & = Y_t z_t^\...
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46 views

Hamiltonian after Co-ordinate Transformation

I have come across this problem in a textbook; $$\dot{x}=ax-bxy$$ $$\dot{y}=-cy+dxy$$ I am asked to show that the transformation $(x,y)\to(p,q)$ where $p=\ln{x}$ and $q=\ln{y}$ leads to a Hamiltonian ...
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28 views

Calculating Hamiltonian of a grid of particles given their spin

I need to calculate the hamiltonian of a grid of particles given their spin. I am given the grid of particles with their spin as well as the values of J and B. The way to find that value is depicted ...
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90 views

Pushforward of Hamiltonian vector field by (reverse) Hamiltonian flow is Hamiltonian

Let $M$ be a smooth manifold with a Poisson tensor $\pi$. Let $X_f = \pi^{\#}(df)$, $X_g=\pi^{\#}(dg)$ be two Hamiltonian vector fields. Let $\Phi_g^u:M\to M$ be the time-$u$ flow of $X_g$. (That ...
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31 views

The flow generated by integral of motion sends orbits of Hamiltonian into orbit of Hamiltonian?

I read these two statement in the notes of my teacher that seem to me opposing. Let $H$ an Hamiltonian and let $\Phi$ an integral of motion of $H$, so that $\Phi$ keeps constant value along the orbit ...
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61 views

Generating function for canonical transformation

I've been reading through some notes on integrable systems/Hamiltonian dynamics, and got stuck on a problem relating to generating functions and canonical transformations. ...
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89 views

Hamiltonian Perturbation Finding First Order Correction

Consider the Harmonic Oscillator as a Hamiltonian System on phase space with Hamiltonian $H = x^2/2 + \rho^2/2$. Now modify the system by adding a perturbation. $H(\epsilon) = x^2/2 + \rho^2/2 + \...
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22 views

Find equilibrium points for Hamilton system.

Here is the given system: $$\begin{cases}x'=x^2+y^2-6 \\ y'=y-x^2 \end{cases}$$ Adding both equations I get: $y^2+y-6=0 \Rightarrow (y-2)(y+3)=0 \Rightarrow y_1=2, y_3=-3, \text{ from there } x_{1,2}=...
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37 views

Extracting the Hamiltonian from given canonical equations

Say I have a physical system that gives me directly a set of equations in terms of the generalised coordinates ($q_{i}$) and generalised momenta ($p_{i}$) as $$\frac{dq_{i}}{dt}=\dot{q_{i}}=f_{1}(\...
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34 views

Ladder Operators for this Hamiltonian $\widehat{H}$

how to find the ladder operators for this hamiltonian: $$\widehat{H}=a\widehat{A}^2 + b\widehat{B}^2$$ where $a$ and $b$ are two real and positive constants. And how to write the hamiltonian in ...
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Generating functions of symplectomorphisms

In physics generating functions are used to produce symplectomorphisms between special symplectic manifolds. More explicitely, given two manifold $M$ and $N$ with respective local coordinates $q_M$ ...
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27 views

optimal control constant

I have the following problem: $$ maximize \int_{0}^{T} [\rho \alpha s(t)x(t) - c(x(t))]dt $$ subject to $\dot{s}(t) = -\alpha s(t)x(t), s(0) = s_{0}, s(T) = s_{f}$ where $\rho, \alpha \...
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47 views

Questions about a proof

I am trying to understand a proof. The statement is The proof begins with: I have some questions about it: ANSWERED Since I am Italian, when it is said "the supreme is ATTAINED" what does it mean? ...
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1answer
68 views

Proof of KAM Theorem

I've been looking for some well-written proof on the KAM Theorem that has all the details and ideas (and if it's not to much that is in some sense elementary). Does anyone happen to know any good and ...
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1answer
52 views

Diagonal subgroup of the Unitary group

Reading Dynamical Systems IV: Symplectic Geometry and its Applications by Arnol'd and Novikov I found the following assertion: The $n$-dimensional monodromy operator of a Hamiltonian system[..]lies ...
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Flow of Hamiltonian vector fields, time dependent flow

I have trouble understanding the notion of time dependent flows of Hamiltonian vector fields: Let $(M, \omega)$ be a symplectic manifold, $H:M \rightarrow M$ a Hamiltonian function. question: In my ...
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Trace of high dimensional matrix multiplication

I am examining when the following expression is true: $$tr( \sum_{i\in G_1}^{k} {A_iB_iC_iB_i}) - tr(\sum_{i\in G_2}^{l} {A_jB_jC_jB_j}) = 0 \quad \quad \quad \: - (*)$$ where $A_{i},B_{i},C_{i}\...
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37 views

Normal mode representation for periodic boundary conditions

In order to decouple the equation of a chain of coupled harmonic oscillator I need to considerate the normal mode basis, here my Hamiltonian: $$H = \sum_{i=0}^{N-1}\frac{p_i^2}{2} + \frac{1}{2}\sum_{i=...
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105 views

Geometric and Visual explanation of Symplectic viewpoint of classical mechanics

I recently learnt that one can express all of classical mechanics on a symplectic manifold. Is it possible to provide an entirely geometric example as to why this is the case? The geometric example ...
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Justification for Legendre transformations (Lagrangian -> Hamiltonian)

In standard Mechanics textbooks Hamiltonian is introduced in a very ad hoc way: Let's try these Legendre transformations and see what happens. Wow, the equations turned out to be more symmetric. How ...
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22 views

Quasi-convexity implies isoenergetically non degeneration?

I'm having some troubles to prove that a quasi-convex function is also iso-energetically non degenerate. In particular, let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ an analytic function, and denote by $...
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Unique Uniformily Viscosity Solution

Say we have the following pocess: $$dX^{\epsilon} = b(X^{\epsilon}(s))ds + \sqrt\epsilon \sigma(X^{\epsilon}(s))dW(s)$$ for $s \in [0, T]$ and $X^{\epsilon}(0) = x_0$ We want to estimate a quantity ...
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68 views

Hamiltonian rigid dynamics

Consider a single free particle of mass $m,$ moving in space under no forces. If the particle starts from the origin at $t=0$ and reaches the position $(x,y,z)$ at time $t,$ find Hamilton's ...
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381 views

Hausdorff dimension of Hamiltonian orbit closure and symplectic leaves

Let $\dot{x} = \Pi \cdot \nabla H$ be a smooth Hamiltonian-Poisson system on $\mathbb{R}^n$. $H: \mathbb{R}^n \to \mathbb{R}$ is the Hamiltonian and $\Pi = (\Pi^{ij})$ is a skew-symmetric matrix of ...
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109 views

Modified Hamiltonian in symplectic Euler method

Now I consider the harmonic oscillator problem. The ordinal differential equation is \begin{align*} \dot{q} &= p \\ \dot{p} &= -q \end{align*} In symplectic Euler method, where \begin{align*} ...
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Confusion about calculation involving E-L and Hamilton equations

As you may know, the E-L equations are given by $\frac{d}{dt} \left( \frac{\partial L}{\partial v_j} \right) = \frac{\partial L}{\partial x_j},\forall j\in\{ 1,\ldots ,n \}$, where $ \begin{matrix} L:&...
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1answer
112 views

Verifiy if a vector field is Hamiltonian with respect to the standard symplectic form

I have the following vector field defined over the manifold $M=\mathbb{R}^2 -(0,0)$: $X(x,y) = \frac{x}{x^2+y^2}\partial _x + \frac{y}{x^2+y^2}\partial _y.$ I have found that $f(x,y) = \tan ^{-1} (y/...
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1answer
71 views

Ehrenfest Theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
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Hamiltonian Flows and Heisenberg picture of quantum mechanics

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial) In particular what I know is that: ...
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34 views

Hamiltonian system and algebraic relations between variables

Consider an Hamiltonian function $$ H(q_1,q_2,p_1,p_2)=p_1\, F_1(q_1,q_2) + p_2\, F_2(q_1,q_2). $$ Assume $q_2=g(q_1)$ for some function $g$. I am interested in seeing what does this property imply ...
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Volume preservation Liouville's Theorem , explanation of proof

I am trying to understand the following proof of Liouville's Theorem , that states that trajectories generated by Hamiltonian equations are volume preserving.The proof can be found in the following ...
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1answer
51 views

The reason for symplectomorphism to conserve the canonical form of the Hamilton equations.

If I have $(M,\omega)$ with Hamiltonian a symplectic manifold, let $(q_1,p_1,...,q_n,p_n)$ be the Darboux coordinates. With these coordinates, the integral curves of the Hamiltonian vector field ...
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1answer
54 views

When are phase space flows induced by the Hamilton equations homeomorphisms?

Suppose I have a certain Hamiltonian of a system, with the corresponding Hamilton equations. The equations induce a certain flow in phase space. Since each point in space has a certain trajectory ...
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125 views

On the existence and uniqueness of solutions of Hamiltonian differential equations

Let $(M,\omega)$ be a symplectic manifold, and $H : M \times [0,1] \to \mathbb{R}$ be a smooth time-dependent Hamiltonian on $M$. Then non degeneracy of $\omega$ implies the existence of a time-...
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1answer
97 views

Constants of motion for a system of two points moving on spherical surface with a force depending only on their relative distance

Consider two points that moves only on a spherical surface of radius $R$. There is only a force between them that has a potential $U(d)$ where $d$ is the distance between the two points. What is the ...
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2answers
198 views

When is the Hamiltonian constant?

In the context of optimal control, when is the Hamiltonian constant? I know that, generally, when $H$ is not explicitly a function of time, it is going to be a constant, but I just did a problem ...
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110 views

Prove existence of a Heteroclinic Orbit

How would you go around proving that there exists a heteroclinic orbit between two equilibria ( in the problem I'm trying to solve, one is a stable node(say n) and the other is a saddle (say s))? I ...