# Questions tagged [hamilton-equations]

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

185 questions
Filter by
Sorted by
Tagged with
27 views

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?...
16 views

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=-...
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 ...
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 ...
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, $\gamma$ ...
16 views

### 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 ...
18 views

### 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 $\sigma^2(Y)$ ?...
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 ...
68 views

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 ...
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 ...
35 views

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 ...
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 ...
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 ...
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+...
31 views

### 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 ...
24 views

### 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 ...