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