Questions tagged [hamilton-equations]

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

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Vector field associated to Hamilton's equation

If we have a Hamiltonian $$ H(x_1,\ldots,x_n;p_1,\ldots,p_n) $$ and Hamilton's equations $$ \frac{d x_i}{dt}=\frac{\partial H}{\partial p_i},\quad\frac{dp_i}{dt}=-\frac{\partial H}{\partial x_i}\tag{1}...
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49 views

Integrate ODE with Leapfrog

$f:\mathbb{R}^n\to\mathbb{R}$ is a function with gradient $g(x)$ and Hessian $H(x)$. My initial conditions are $(x_0, v_0)$. I would like to discretize this system of ODEs $$ \begin{align} \dot x(t) &...
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24 views

Pontryagin principle for fuel minimization of moon lander: is it the same for time minimization also?

I am checking this document that applies Pontryagin principle to solve moon lander fuel's optimization problem https://rc.library.uta.edu/uta-ir/bitstream/handle/10106/23955/Ocampo_uta_2502M_12322.pdf?...
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Is this solution in stochastic optimal control problem equivalent?

Consider a stochastic optimal problem, where $$\mathrm dx_t = b(t,x_t,u_t)\,\mathrm dt + \sigma (t,x_t,u_t)\,\mathrm dB_t, \qquad x_0=x,$$ with cost functional $$J(u) = E \left[ \int_0^T f(t,x_t,u_t)\,...
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33 views

How to compute the Hamiltonian easily

I am tasked to find the Hamiltonian of a system which comprises of a body of mass $m$ that moves on the surface defined by $z = x(x-1) + y(y-1)$. I have computed the Lagrangian to this problem and ...
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39 views

Canonical transformation and generating function

I am stuck on an exercise of which I do not understand the solution. The exercise is the following: Consider the Hamiltonian system $$H(\theta, p, t) = \dfrac{(p-\omega t)^2}{2} - k\cos(\theta)$$ ...
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82 views

Orbital resonances - expansion of disturbing function

I want to study the orbital resonance type $3:1$ between an asteroid and Jupiter. For this purpose, I found the expansion of the disturbing function in Celletti A., Stability and Chaos in Celestial ...
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1answer
50 views

Sphere is a symplectic submanifold of Lie algebra $S0(3)^*$

I'm working through this example in Peter Olver's textbook Application of Lie Groups to Differential equations and I am having some trouble and was wondering if somebody could point out where I went ...
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1answer
39 views

Rotations in $R^2$ are Poisson maps

So I am trying to learn about Hamiltonian systems from Peter Olver's book, applications of Lie groups to differential equations. Right now I am on the section talking about Poisson maps and there is ...
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1answer
31 views

Finding a Hamiltonian function with or without Lagrangian

Is it possible to find a Hamiltonian function for a system without having the Lagrangian first? I have the pendulum equation without damping and without sinusoidal driving force, that is $$\frac{d^2\...
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30 views

Hamilton's function according to the Delaunay variables

Consider the following system: \begin{equation} \ddot{x}_1=-\frac{\mu_{\oplus} x_1}{r^3}-\frac{\mu_\oplus R_{\oplus}^2J_2}{r^5}\bigg(\frac{3}{2}x_1-\frac{15}{2}\frac{x_1x_3^2}{r^2}\bigg)\\ \ddot{x}_2=-...
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1answer
83 views

Integrating $\int \sqrt{2me-mkr^2-\frac{1}{2}m br^4 - \frac{a^2}{r^2}} \,dr$

I was trying to find Hamiltons principle function, $S$, for the Hamiltonian: $$ H = \frac{1}{2} m \left( P_{r}^2 + \frac{P_{\theta}^2}{r^2} \right) + \frac{1}{2}kr^2 + \frac{1}{4} b r^4$$ After ...
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1answer
53 views

Calculate the Poisson bracket {A, H}, and check if there is a value of c for which A is a constant of motion?

Consider the Hamiltonian H given by$$H=(x,y,z,p_x,p_y,p_z)= \frac{p^2_x}{2m}+\frac{p^2_y}{2m}+\frac{p^2_z}{2m}-\frac{1}{\sqrt{x^2+y^2+z^2}}$$ where x(t), y(t) and z(t) give the location of a particle ...
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55 views

Overdamped vs Underdamped Langevin

Consider the : $\textbf{Underdamped Langevin}$ \begin{align} dX_t&=V_tdt \\ \frac{m}{\gamma}dV_t&=-V_tdt-\nabla \phi(X_t)dt+\sqrt{2D} W_t. \end{align} I believe $m$ is the mass, $\gamma$ ...
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How to use Euler Theorem on homogeneous function to obtain $V' = -[2T - (\frac{\partial T}{\partial q}|q) - (\frac{\partial \pi}{\partial q}|q)]H$

Given the expression $V' = -[-(q|\frac{\partial H}{\partial q}) + (\frac{\partial H}{\partial p}|p)]H$ where (a|b) denotes the scalar product, and H is the hamiltonian ($H = T(q, p) + \pi(q)$, and ...
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Variance of random variable whose support is contained in the support of another random variable.

Suppose we have a random variable $X$ defined on the set $A$. Let $Y$ another random variable, with the support $B\subset A$. What can I say on the relation between $\sigma^2(X)$ and $\sigma^2(Y)$ ?...
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60 views

Constants of motion for Lagrangian $\frac{1}{2}m((\dot{x}-wy)^2+(\dot{y}+wx)+\dot{z}^2)$

Consider a particle of constant mass $m$ with Lagrangian $$L (x, y, z, \dot{x}, \dot{y}, \dot{z}) = \frac{1}{2}m((\dot{x}-wy)^2+(\dot{y}+wx)+\dot{z}^2)$$ where $(x(t), y(t), z(t))$ is the location of ...
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1answer
68 views

Hamilton equations-Symplectic Euler method

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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1answer
47 views

Poincaré-Bendixson theorem and the Sinai Billiard

One of the conclusion of the Poincaré-Bendixson Theorem is that in planar dynamical systems chaotic motion could not arise. But in the Sinai Billiard trajectories in fact are chaotic...how is this ...
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Why does $H_{y'}(x,y,y',p) = 0$ and $H_{y'y'}(x,y,y',0) \leq 0 $ imply $y$ is an optimal curve?

I am reading Liberzon's Calculus of Variations and Optimal Control theory, and trying to understand the Maximum principle. It is currently going over the Hamiltonian formulation. I will define some of ...
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24 views

When can we interchange operations involving partial derivative and integral, and how does Hamiltonian formulation affect that

Say we have an integral $$ B=\int dt \frac{\partial}{\partial q} f(t,q,\dot{q}), $$ where $t$ is an independent parameter, $q=q(t)$ is a dependent variable, $\dot{q}=dq/dt$ and $f$ is an arbitrary ...
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51 views

Generating functions for the iterates of a symplectomorphism

$\newcommand{\coloneqq}{\colon\!=}$ In an exercise at the end of section "5.1 Periodic points" of Ana Cannas da Silva's book "Lectures on Symplectic Geometry" it is stated that, ...
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34 views

Understanding a passage involving total and partial derivatives

I'm reading Michael Betancourt's "A Conceptual Introduction to Hamiltonian Monte Carlo". However my question is just about a particular passage involving total and partial derivatives. On ...
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122 views

Hamiltonian set of equations

I'm learning differential equations and trying to solve the following problem: $x_1'=x_2^2$ $x_2'=x_4+x_3$ $x_3'=x_4^2$ $x_4'=x_1-x_2+x_3^2$ We want to determine whether the system of equations is ...
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1answer
108 views

Hamilton equations by flow of Hamiltonian vector field

I am working on an exercise for my differential geometry class and would like to know if my solution to a problem is correct. The problem is as follows: A symplectic manifold $S$ is - by definition - ...
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34 views

Reference For Hamiltonian Manifolds/ Hamiltonian structures

I'm trying to learn about Hamiltonian Structures of Partial differential equations using Peter Olver's Book 'Applications of Lie Groups to Differential Equations', but Olver uses the notation of a ...
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1answer
23 views

Why does the Hamiltonian vanish for $\int f(x,y)\sqrt{\dot{x}^2+\dot{y}^2} \text{ d}{t}$?

So I have this integral $$ \int_{t_1}^{t_2} f(x,y)\sqrt{\dot{x}^2+\dot{y}^2} \text{ d}t, $$ where $x,y$ are functions of $t$ and $f$ is a function of just $x,y$. Also $\dot{x}$ denotes the first ...
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2answers
90 views

Non-unique solutions for ODE with boundary conditions at infinity

Ello, I am looking for solutions to equations such as $$ U ^ \prime (y) - y^{\prime \prime} + \beta y^{\textit{IV}} = 0$$ where $\beta >0$ is a constant and $U(x)$ is a function whose Taylor ...
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0answers
82 views

Asymptotic of isospectral matrix flow, for $t\to\infty$

I have an isospectral flow on $\mathfrak{su}(N)$, for some $N>1$ of the form: $$ \dot{W}=\left[ LW,W\right], $$ where $W\in\mathfrak{su}(N)$ and, introducing the Frobenius inner product $\langle A,...
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2answers
86 views

Noether‘s Theorem and Moment Maps

Noether‘s Theorem says that every continuous symmetry of a physical system (i.e., a Lie group action on phase space ${\bf R}^{2n}$ preserving a Hamiltonian $H$) leads to a conservation law (i.e., a ...
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1answer
72 views

Hamiltonian - Classical Mechanics

I'm studying classical mechanics reading Mathematical Methods of classical mechanics, by Arnold and doing some exercises lists. As I'm studyng by myself, I got stuck on this exercise. Consider the ...
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1answer
35 views

ResNet derivative issue wrt the Hamiltonian

I'm reading Deep Learning as OCP and I've got some doubts about a derivative. Consider an Hamiltonian built in the following way $$\mathcal{H}(y,p,u) = \langle p,f(Ky+\beta)\rangle$$ where $K\in\...
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1answer
225 views

How to show the Hamiltonian is conserved

I have the langrangian $L = \frac 1 2m((1+4r^2)r’^2) + \frac 1 2 mr^2 θ’^2 -mgr^2 $ I worked out theHamiltonian and 4 Hamiltonian equations and got $\frac {dpθ}{dt} = 0 $ for one of them $pθ$ is the ...
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One-Parameter Group $F:\mathbf{R}\to\text{End}(C^\infty(\mathbf{R}^{2n}))$ Not Generated By Some $\bar F:\mathbf{R}\to\text{End}(\mathbf{R}^{2n})$

Reading this article, and the definition of one-parameter group, I wonder what naturally occuring (in physics or elsewhere) one-parameter groups $\{F_t:C^\infty(\mathbf{R}^{2n})\to C^\infty(\mathbf{R}^...
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1answer
60 views

Compactly supported hamiltonian diffeomorphisms

I'm having trouble with an Exercise (12.3.6) from McDuff's and Solomon's book, "Introduction to Symplectic Topology" (3rd Ed.). The goal is to prove the monotonicity of the symplectic ...
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1answer
99 views

Hamiltonians, PDE, Density of a Fluid.

I'm struggling with Hamiltonians and PDE. My question is loosely based on the PDE of this paper by Figalli, Gangbo and Yolcu. Before asking a question, let me spell out some prior knowledge, coming ...
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18 views

Different invariant tori in the case of a 2D harmonic oscillator

Both textbooks I'm currently reading (Mathematical Methods of Classical Mechanics by Arnol'd - get it here - and Classical Mechanics by Shapiro - here) say that, in the case of the planar harmonic ...
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35 views

Solve problem with bounded control

max integral of (e^(-rt))*(1-u)xdt from 0 to T where x˙=ux, u(t)∈[0,1] and x(0) = x0 greater than 0. I wrote the current value Hamiltonian and Lagrangian, optimality conditions etc. Where Lu = -x+mx+...
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Calculus of variations for a Lagrangian functional

I'm trying to understand eq. (2.2) in the following excerpt taken from this lecture notes (p. 10): I'm sure that this result is rather trivial, but I've got a hard time to follow the notation. Since ...
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Find H for a particle in an infinite well

So I have this problem: 1- A particle in an infinite square well has a wave function that is $\psi(x,0)=A[\psi_1(x)+\psi_2(x)] $ with $\psi_n$ the n-th steady state. a)normalize $\psi(x,0)$ I already ...
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13 views

Resonant and non-resonant tori density in non-degenerate system

I'm following the discussion on the page 290 of Mathematical Methods of Classical Mechanics by V. I. Arnol'd (you can download it here), and I've encountered the fact that in a nondegenerate system, ...
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18 views

Uniqueness of momentum in Hamiltonian mechanics

Hamilton's equations of motion are \begin{align} \dot{q} = \nabla_p H(q, p) ~~~~~~~~~~~~~~~~~~~~~ \dot{p} = -\nabla_q H(q, p). \end{align} Suppose that at a particular $q = q_0$, I know $\dot{q} = \...
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26 views

How can we prove Liouville's theorem (Hamiltonian) in a matrix form?

I try to prove a dispersed form of Liouville's theorem, and I transform phase space to: $$ \mathrm{d}X\mathrm{d}P = |J(t;0)|\ \mathrm{d}x\mathrm{d}p $$ so, we just need prove: $$ J(t;0) = |\frac{\...
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How to use Noether’s theorem in the hamiltonian formulation of mechanics?

Given that there is a constant of motion $f$, ie a function $f$ that Poisson-commutes with the Hamiltonian, how can you guess (or generate) a canonical transformation to new coordinates such that one ...
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1answer
38 views

partial derivative of the Hamiltonian inside a function

Let $f( \mathcal{H}(q,p))$ where $q(t)$ and $p(t)$ are time dependent. My Professor wrote: $$ \frac{\partial f}{\partial t} = f' \frac{\partial H}{\partial t} $$$$ \frac{\partial f}{\partial q} = f' \...
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24 views

Different definitions of Hamiltonians

I've found two different definitions of the Hamiltonian functions as related to a Lagrangian $\mathcal{L}(t,q,\dot{q})$. $$H(t,q,p):=\sup_{\dot{q}}\big(p_j\dot{q}^j-\mathcal{L}(t,q,\dot{q})\big);$$ $$...
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1answer
38 views

Equality from Legendre transformation

Let $Q$ be a manifold. I recall that for a $\dot{q}$-uniformly convex Lagrangian $\mathcal{L}:\mathbb{R}\times TQ\rightarrow\mathbb{R}$, the Legendre transformation \begin{equation*} \begin{array}{...
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20 views

Proving that a “point transformation” is always a canonical transformation, even for multiple degrees of freedom

In the context of classical mechanics and canonical transformations, here denoted generally as $(q,p) \rightarrow (Q,P)$, a point transformation (as opposed to the more-general "contact ...
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22 views

Angle-Action variables for 2D or 3D conservative systems

For a 1D conservative Hamiltonian like $H(q,p)$, we can transform to Angle-Action (AA) variables using the following formulae, which are based on area-conservation of this canonical transformation (...
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1answer
150 views

Geodesic flow are generated by Hamiltonian vector field

Let $(M,g)$ be a Riemannian manifold and consider the Hamiltonian \begin{equation*} \begin{array}{rcl} H:T^*M & \rightarrow&\mathbb{R} \\ (q,p) & \mapsto&H(q,p):=\...