# Questions tagged [hamilton-equations]

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

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### Stability of Hamiltonian system on degenerate critical point.

I'm trying to find information on the stability of the following ODE: $$x'' = x^4-x^2.$$ We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
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Consider the following deterministic optimisation problem \begin{align} J(t) = \min_{c(t)} \ & \frac{1}{2} \int_0^\infty e^{-\delta u} \left( x(u)^2 + \lambda y(u)^2 \right) du \\ s.t. \ &c(t) ...
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### Conversion from cartesian coordinates to generalized coordinates

Say we have a system with two particles with mass $m_1=m$ and $m_2=m$ with positions described in cartesian coords. by $\mathbf r_1=(x_1=0, y_1=C-q_1)$ and $\mathbf r_2=(x_2=q_1+q_2, y_2=0)$. Its ...
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### $\frac{d}{dt}(d\psi^t)_x(Z)=(d\psi^t)_x([X_H,Z])$ for a critical point $x$ of a Hamiltonian $H$

Let $W$ be a closed symplectic manifold, $H:W\to \Bbb R$ a Hamiltonian, $x\in W$ a critical point of $H$, $X_H$ the associated Hamiltonian vector field, and $\psi^t:W\to W$ the (global) flow of $X_H$. ...
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### Transform a differential equation into Hamiltonian form

I am currently doing the following exercise in the book Modern Geometry - Methods and Applications Part I by Dubrovin, Fomenko, and Novikov. Exercise 33.4.1: Consider the differential equation \begin{...
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### Hamiltonians and symplectic transformations

Let $H:\mathbb{R}^{2n}\rightarrow \mathbb{R}$ be a Hamiltonian function, $\phi:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$ be a symplectic transformation, that is $(D_x\phi(x)) J (D_x\phi(x))^T = J$ ...
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### When do skew symmetric and symmetric positive definite matricies commute?

Let $J$ be a skew-symmetric matrix and $A$ be a symmetric positive definite matrix. When is it possible for $JA = AJ$?
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### Can the tangent and cotangent bundles have the same coordinates? Symplectic form in cotangent coordinates

This is somewhat related to another question I asked at this link. In classical mechanics, the configuration of some mechanical systems (say, a double pendulum) can be described by a point on an n-...
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