Questions tagged [hamilton-equations]

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

2
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1answer
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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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0answers
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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\...
3
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0answers
33 views

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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0answers
30 views

Symplectic map and Poisson bracket notation

I'm having trouble trying to work with a certain notation. Def: A diffeomorphism $\Phi$ of $\mathbb{R}^{2n}$ is symplectic if, for all $f,g\in C^{\infty}(\mathbb{R}^{2n})$, $$\{f\circ\Phi,g\circ\Phi\}...
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31 views

Total and partial derivate of Hamiltonian

I want to show that the total and partial derivative regarding the time t of the Hamiltionian are equal. Doing following calculation: \begin{align} \begin{split} &\dfrac{d}{dt}H(x_*(t),p_*(...
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30 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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14 views

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 ...
1
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1answer
35 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 ...
2
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1answer
41 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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33 views

Hamiltonian mechanics is good because of the symplectic structure of Hamiltonian systems.

I was reading the wiki on Hamiltonian mechanics, and I stumbled across the moto : The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic ...
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1answer
44 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-...
2
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1answer
84 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
40 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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0answers
59 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 ...
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0answers
33 views

Trajectories in phase space of a non autonomous Hamiltonian

If I have some autonomus hamiltonian, then it is a constant of the motion, and all the trajectories will lie on constant set levels. For example, for the double well potential problem $$H(x,p) = p^...
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26 views

What is proof of direction of Change in position vector $d \vec r$

How can I prove that the direction of an infinitesimal change in position vector $\mathrm d\vec r$ is the same as that of the instantaneous velocity $ \vec v=\mathrm{d}\vec r/\mathrm{d}t$? What I ...
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1answer
39 views

Hamiltonian system and exponential map — backward

Consider a linear ODE $$ \dot x(t) = A\,x(t). $$ A solution is $x(t)=\exp(t\,A)$ where $\exp$ is defined by $$ \exp(A) = \sum_{n=1}^n\frac{A^n}{n!}. $$ Consider a Hamiltonian system $$ \dot x(t) = \...
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Momentum constraints for a Singular Lagrangian

Note I've explicitly indicated it at points in this question, but unless stated otherwise $i,j,k \in \{1, \ldots, n\}$, $a,b,c \in \{1, \ldots, R_W\}$, and $\alpha, \beta, \gamma \in \{R_W + 1, \...
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0answers
28 views

Dirac measure on periodic orbits for Hamiltonian Dynamcis

An iterative map $T: x_{n+1} = T(x_n)$, given an initial state $x$, defines a periodic orbit: $O_x=\lbrace x, S(x), ...,S^{n-1}(x) \rbrace$ , with $S^n(x) = x$. The periodic orbit supports a measure ...
5
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1answer
115 views

Intuition about Poisson bracket

I've been reading about Hamiltonian mechanics which in its mathematical description uses Poisson manifolds From my limited understanding, on a Poisson manifold $M$ we can look at the Poisson bracket ...
0
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1answer
61 views

Constant trajectory in Hamiltonian system

Let the (non-canonical) Hamiltonian system be given in the form $$\dot{x}=J(x)DH(x)$$ where $H(x)$ is the Hamiltonian function, $J(x)$ is the skew-symmetric symplectic matrix, and $DH$ denotes the ...
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2answers
66 views

Undamped simple Pendulum written in Hamiltonian form

Show that the equation of motion for an undamped simple pendulum $y'' + \frac{g}{l}sin(y)=0$ can be written in Hamiltonian form. So this is the Euler-Lagrange form of the simple pendulum equation,...
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1answer
26 views

Show that $H(v, y) = \frac{v^2}{2} - F(y)$ is a first integral

$H(v, y) = \frac{v^2}{2} - F(y)$ is a hamiltonian of $y' = v, v' = f(y)$ Linear first integrals are of the form $I(x) = b^Tx + c$ where $b \in ℝ^d $ and $ c \in ℝ$ Quadratic first integrals are of ...
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0answers
29 views

Relativistic Hamiltonian with Full Mass Matrix

In some application of Hamiltonian Monte Carlo one can provide a full mass matrix (metric tensor) for the kinetic energy in Hamiltonian equations: $$ K(p) = \frac{1}{2}p^TM^{-1}p$$ which reduces to ...
1
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1answer
41 views

optimal control to minimise a path

I'm having issues solving this problem. Here is what I have tried so far. $$ u=\dot {x_1} + x_1 $$ $$ J= \frac{1}{2} \int_{0}^{t_1}((2x_1)^2+2\dot {x_1} x_1 + \dot {(x_1)^2})dt$$ Can I proceed and ...
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0answers
106 views

Hamiltonian and Lagrangian correspondence

I'm trying to clarify how we get a Hamiltonian directly from a Lagrangian using the Legendre transform. Let me give some preliminaries for my question to make sense. A Hamiltonian system is a triple ...
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1answer
38 views

What does it mean for a vector field to preserve area?

I was reading a book about hamiltonian mechanics. After computing the divergence of the hamiltonian vector field to be identically zero, the author adds: "...thus the vector field is divergence-free ...
2
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1answer
50 views

Calculus of Variations: Hamilton’s canonical equations

In Calculus of Variations, Hamilton’s canonical equations (Calculus of Variations and Optimal Control Theory by Daniel Liberzon, p. 45) are $$y'~=~H_p,\tag{1}$$ $$p'~=~-H_y.\tag{2}$$ I understand ...
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0answers
81 views

Calculus of Variations. Finding the extremals of a perturbed Lagrangian

Im trying to solve the following problem: Approximate with an error of $O(\epsilon ^3)$ the extremals of the Lagrangian $$L(y,y',x) = y^2 + (y')^2 - 2y \sin(x) + \epsilon y^3$$ with $y(0)=1$ ...
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0answers
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Markov Property fos Ising type Models

We are interested in proving the Markov property for the long range Ising type model in $\mathbb{Z}^d$. Setting: Define $\Omega = \{-1, +1\}^{\mathbb{Z}^d}$ the space of all possible configurations ...
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1answer
111 views

Proof for Liouville's theorem - Hamiltonian mechanics

Studying analytical mechanics I encountered Liouville's theorem which states: In phase spase, the Hamiltonian flow preserves volumes The book I'm studying is Analytical Mechanics - A. Fasano, S. ...
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1answer
52 views

How to show nonlinear Schrodinger equation is an infinite dimensional Hamiltonian system?

The nonlinear Schrodinger equation is $$ ih\frac{\partial \psi}{\partial t} = -\frac{h^2}{2}\Delta \psi + V\psi-|\psi|^{p-1}\psi $$ From Wiki, I know the Hamiltonian system is ...
0
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1answer
78 views

Are Hamiltonian trajectories geodesics on the cotangent bundle?

Suppose we have a Hamiltonian dynamics on a phase space, whose base space is also a Riemannian manifold. I was wondering if the Hamiltonian trajectories are whether geodesics or only locally geodesics ...
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0answers
29 views

Finding action-angle variables for integrable maps

Suppose I have an Liouville-Arnold integrable area preserving map $(\bar{x}, \bar{y}) = F(x,y)$ with a first integral $I(x,y)$. How does one seek appropriate action-angle variables such that the map ...
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0answers
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Poisson maps in Hamiltonian PDEs (KdV in particular)

I've run into a bit of a sticky point in doing some background research about the Hamiltonian structure of KdV. What I know is: A Poisson bracket is written in coordinates $x$ as $$\{F, G\} = (\...
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0answers
72 views

How do I find canonical coordinates for the Lorentz group generators?

Consider the Poisson brackets (symplectic structure) given by the Lorentz algebra (Lie algebra of $SO(1,3)$) $$\{M^{AB},M^{CD}\} \equiv \omega_{AB,CD}\mathrm{d}M^{AB} \mathrm{d} M^{CD} = \eta^{AC} M^{...
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0answers
56 views

The trajectory of $(\dot q(t),\dot p(t))=(p(t),-q(t)^3+\sin t)$ is bounded

The original question is to show that for small $\varepsilon>0$, all solution of ODE $(\dot q(t),\dot p(t))=(p(t),-q(t)^3+\varepsilon\sin t)$ is bounded. Using KAM theory we can show that for a ...
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1answer
47 views

Non-Hamiltonian systems of odes on a plane and stability of their equilibria

For a Hamiltonian system of odes on a plane, the eigenvalues of the linearisation matrix of fixed points are of the form $\pm \lambda$ for $\lambda \in \mathbb{R}$, (hyperbolic) or $\pm i \mu$ (...
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2answers
156 views

Einstein's convention and Hamilton's equations in $\Bbb R^3$.

Consider $F = -\nabla U$ a conservative force field in $\Bbb R^3$. Assume we describe the motion of a unit mass particle under this force field by a curve $q(t) = (q^1(t), q^2(t), q^3(t))$. We have ...
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0answers
48 views

Finding action-angle variables for integrable Hamiltonian

How to introduce action-angle variables in the following integrable 2 d.o.f. Hamiltonian system? $$H(q,p,x,y) = \frac{y^{2}}{2} - \frac{x^{2}}{2}\left(p^{2} + \omega^{2}q^{2}\right) + \frac{x^{4}}{4}$...
7
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1answer
205 views

Proving that system is Hamiltonian

I am trying to show that the PDEs governing stratified flow are Hamiltonian. The approach is based on the paper "Nonlinear Stability Analysis of Stratified Fluid Equilibria which can be found here ...
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1answer
154 views

Do the Euler Lagrange equations hold meaning for an infinite action? [closed]

This question is edited and migrated from mathstackexchange The Euler–Lagrange equation is an equation satisfied by a function $q$, which is a stationary point of the functional $S(\boldsymbol q) = ...
4
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1answer
186 views

Dynamical System is fixed point at origin hyperbolic or asymptotically stable and is the system Hamiltonian

I am given a system $\dot x = f(x,y) = (x^2 + y^2)(x^3 + y^2x -2y - x) \\ \dot y = g(x,y) = (x^2 + y^2)(y^3 + x^2y +2x - y) $ and I am asked if the fixed point at $(0,0)$ is hyperbolic or ...
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1answer
78 views

What is a symplectic form of the rotation group SO(n)

I need to prove that the Hamiltonian system of the rigid body motion $$ \begin{cases} \dot{R}_t=P_tJ^{-1},\\ \dot{P}_t=2R_t\Lambda,\quad\text{$\Lambda$ is the Lagrange multiplier}\\ R_t^T R_t-I=0, \...
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0answers
199 views

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

Starting with a closed, connected Riemannian manifold $(M^n, g)$, using the "product Euclidean metric" to form an associated Riemannian metric $\tilde{g}(p,v)((v_1,w_1),(v_2,w_2))) =$ $g(p)(v_1,v_2) + ...
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2answers
109 views

Find the Hamilton's equations

Consider the functional given by $$ \ \large \ J(y)=\int_{a}^{b} \sqrt{(t^2+y^2) (1+\dot y^2) } \ dt .$$ Find the Hamilton's equations . Answer: I am unable to find the Hamilton from the ...
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1answer
106 views

Gauge invariance of the Hamiltonian

Consider a Lagrangian $L(x,\dot x,t)$ and a corresponding Hamiltonian $H=\dot xp-L$ where $p=\partial L/\partial \dot x$ which satisfies Hamilton's equations $$\frac{\partial H}{\partial x}=-\dot p$$ $...
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0answers
106 views

How to find a Lax pair

I've been getting into Hamiltonian PDE's lately and they give the KdV equation a lax pair to proof that it is integrable in some sense. My question is the following how does on find a lax-pair if we ...
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2answers
91 views

Hamiltonian from Lagrangian $L= \frac{m}{2}(\dot{r}^2+r^2\dot{\theta^2})+ \frac{k\cos(\theta)}{r^2}$

I'm doing the first exercises with the Lagrangians and Hamiltonians. Let: $$L= \frac{m}{2}(\dot{r}^2+r^2\dot{\theta^2})+ \frac{k\cos(\theta)}{r^2}$$ $$p_1=m\dot{r}$$ $$p_2=mr^2\dot{\theta}$$ $$H=\...
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0answers
69 views

Action-angle variables in a 2 degree of freedom integrable Hamiltonian?

Suppose I have the following Hamiltonian $$H(q, p, x,y) = \frac{p^{2}}{2} - \frac{q^{2}}{2} \left(\left(\frac{y^{2}}{2} + \omega^{2}x^{2}\right)^{2} - \frac{q^{2}}{2}\right)$$ Where $(q, p, x,y) \in \...