Questions tagged [hamilton-equations]
Use this tag for questions related to Hamilton's equations.
220
questions
2
votes
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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 ...
- 327
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0
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33
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Wrong sign in co-state of optimal control problem
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) ...
- 341
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1
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How to verify positive definitiveness of the given Kinetic term?
I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12:
$$\frac{\mathcal{S}}{\mathcal{T}}= \int dt\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)+11.3c_0^...
- 121
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38
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Why can we interpret forces on particles as components of a smooth convector field in the n-Body problem?
I'm trying to understand Example 22.18 (The $n$-Body Problem) in John Lee's Smooth Manifolds textbook. I'm confused by the step where the forces in Newton's second law [Eq. (22.12)] are interpreted as ...
- 13
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votes
1
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33
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How to solve simple second order ODE with RHS $x / \sigma^2$
I have the following Hamiltonian systems for $i=1, \ldots, d$ and $\sigma_i > 0$
$$
\begin{align}
\frac{d}{dt} x_i &= v_i \\
\frac{d}{dt} v_i &= \frac{x_i}{\sigma_i^2}
\end{align}
$...
- 159
2
votes
1
answer
57
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Equation of motion for Hamiltonian for n bodies.
Finding the arising equation of motion for given the Hamiltonian of n particles
$$H = \frac{1}{2m} q_n^2+\frac{\alpha}{2}(y_{n+1}-y_n)^2+\frac{\beta}{4}(y_{n+1}-y_n)^4$$
The $\alpha,\beta, m$ are ...
- 391
0
votes
2
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41
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Turn two ODEs first degree into a Hamiltonian with code
This is more like a coding question, but I thought I'll post it here because of its mathematical foundation.
Two ODEs first degree can describe a Hamiltonian System. The connection between the ODEs ...
- 119
2
votes
1
answer
77
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Jacobi identity for Poisson bracket in local coordinates
Suppose a bivector field $\pi^{ij}$ such that $\pi^{ij}=-\pi^{ji}$, $\pi^{ij}\partial_{i}f\partial_{k}g=\{f, g\}$ defines a Poisson bracket $\{,\}$ on a smooth manifold (Einstein's summation is ...
- 1,197
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Is there a theory of "quadratic" Hamiltonian evolutions on Poisson manifolds?
I am dealing with a PDE which can be written in the form
$$\frac{d}{dt} f(t) = \{a, f(t)\} + \{\{b, f(t)\}, f(t)\}$$
A Hamiltonian equation on a Poisson manifold has the following form:
$$\frac{d}{dt} ...
- 121
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votes
0
answers
71
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Does the Hamiltonian system have unbound solutions?
I want to know if it is possible to determine if the following Hamiltonian system has unbound solutions. Let us consider the Hamiltonian function
$$ H(x,y,p_x,p_y) = \frac{1}{2}(p_x^2+p_y^2) + \frac{x^...
- 123
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37
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How can I apply the Hamiltonian function and Pontryagin's maximum principle in the context of Optimal Control Theory?
I am really struggling to grasp how the Hamiltonian Function and Pontryagin's Maximum Principle work in the context of Optimal Control Theory (Maths for Economics) course. I am given the following ...
2
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0
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126
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Relation between symmetries and Lyapunov exponents
Let us consider a system i) that is Hamiltonian, and ii) where we can apply the Oseledets theorem.
The presence of a symmetry ensures the presence of a vanishing (zero) Lyapunov exponent.
To be more ...
- 165
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59
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Canonical transformation for hamiltonian system possessing first integral
Let say I have a Hamiltonian system
$$
\begin{align*}
\dot p &= -H_q \\
\dot q &= H_p
\end{align*}
$$ with Hamiltonian $H(p,q)$, coordinates $q \in \mathbb{R}^2$ and momenta $p \in \mathbb{R}...
- 220
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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 ...
- 2,613
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0
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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 $X_H$. ...
- 1,630