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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 ...
Guybrush's user avatar
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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) ...
NC520's user avatar
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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^...
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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 ...
Maple's user avatar
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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} $...
Physics_Student's user avatar
2 votes
1 answer
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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 ...
unknown's user avatar
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2 answers
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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 ...
Mo711's user avatar
  • 119
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1 answer
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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 ...
Daigaku no Baku's user avatar
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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} ...
Robert Wegner's user avatar
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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^...
alejandro's user avatar
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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 ...
astute-hoplite's user avatar
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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 ...
Doriano Brogioli's user avatar
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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}...
Maksim Surov's user avatar
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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 $(D_x\phi(x)) J (D_x\phi(x))^T = J$ ...
Ben94's user avatar
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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 $JA = AJ$?
cisprague's user avatar
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Choosing correct parameters in a model with hamiltonian equation

I am working on the hamiltonian of a system related to the extension of the Potts Model which is Cellular Potts Model. The total hamiltonian of the system is: $$ H = H_1 + H_2 $$ $$ H_1 = - J \sum_{\...
wallevic's user avatar
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Constructing diagonalizable matricies such that matrix exponential of their product is easy to compute.

Question Consider a real skew-symmetric matrix $J$ and a real symmetric positive semi-definite matrix $A$. We want to parameterize a construction of these matrices so that it is easy to compute $e^{JA}...
cisprague's user avatar
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2 votes
4 answers
120 views

The use of subscripts in L11 E04 - Classical Mechanics by Leonard Susskind

I am struggling to get my head around the soultion to exercise 4 lecture 11 in The Theoretical Minimum (Classical Mechanics). The exercise and its solution can be found here: https://tales.mbivert.com/...
Peter Petrov's user avatar
3 votes
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90 views

What assumption of Noether's theorem fails in this Hamiltonian system with infinitely many particles?

A consequence of Noether's theorem is that the energy of a Hamiltonian system is conserved if and only if the Hamiltonian is time-translation invariant. However, to my surprise I found some ...
Maximal Ideal's user avatar
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unknown manipulation to deal with variation and obtain energy balance equation?

I'm studying a paper relevant to using d'Alembert's principle to describe the motion of fluid. The authors shows an interesting manipulation to obtain energy balance equation, which makes me confused. ...
106207436's user avatar
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67 views

Instability of equilibrium points of system $x'=y^3$, $y'=\cos(x)\sin(x)$

We take a look at the system $$ x'=y^3, \quad y'= \cos(x)\sin(x). $$ It has a Hamiltonian $H(x, y) = \frac{1}{4}y^4 - \frac{\sin(x)^2}{2}$ for $(x, y) \in \mathbb{R}^2$. It is clear that the points $n\...
Hyperbolic PDE friend's user avatar
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1 answer
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Conditions for a vector field to be complete.

If we consider in a symplectic N-dimensional manifold $(M,\omega)$ the infinitely differentiable functions at all points of $M$ denoted by $C^\infty(M,\mathbb{R})$, and a Hamiltonian vector field ...
Felipe Dilho's user avatar
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1 answer
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Reference request: Why is an integrable system called an integrable system and why is the dynamical billiard on a disk completely integrable?

I am seeking detailed reference or references to help me understand the following: Relevant history and motivation behind the term "integrable system" with appropriate primers The meaning ...
Cartesian Bear's user avatar
3 votes
0 answers
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Is there any practical use for octonions? [closed]

Quaternions are useful for describing orientation/ rotations in 3- dimensions, however is there much practical use for an 8-dimensional base hyper complex number id est Octonions?
Olly Doe's user avatar
1 vote
0 answers
299 views

Find the separatrix of a dynamical system

I have the following system with kinetic energy: $T = \frac{m\dot{x}^2}{2}$ and potential energy: $V=ax-x^3$. I want to find if there are any separatrices and if there are, is there a condition on $a$....
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Differential equations when the flow vanishes at nonisolated points

Almost all books on differential equations discuss fixed points where the flow vanishes, i.e., for a system of the following kind $$ \dot{\mathbf{x}} = f(\mathbf{x}) $$ points $\mathbf{x}$ where $f(\...
B215826's user avatar
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0 answers
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Generalized version of Routh's theorem

I am studying the dynamics of vortices in simply connected domains, in other words, in regions that are conformally equivalent to the circle. Then, through the theory of conformal maps, the ...
Júlio César's user avatar
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1 answer
86 views

Hamiltonian vector field vs Heisenberg picture, from a view point of geometric quantization

Let $(M,\omega)$ be a symplectic manifold, and let $X_f$ denote the Hamiltonian vector field for $f\in C^{\infty}(M)$. I know that the integral curve $$(X_H)_{\gamma (t)}=\dot{\gamma}(t)$$ means the ...
s.h's user avatar
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1 vote
1 answer
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Computing Poisson brackets

Suppose I have Hamiltonian :H = $\frac{1}{2}(S_1^2+S_2^2+\beta S_3^2)+R_1$, and two first integrals: $f_1 = R_1^2+R_2^2+R_3^2$,$f_2 = R_1S_1+R_2S_2+R_3S_3$. And also i know how to compute Poisson ...
VadimStacheff's user avatar
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1 answer
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Show that if $ \lim _{|x| \rightarrow \infty} \int_0^x h(s) d s=+\infty $ then all solutions to this ODE are bounded.

a) Analyze $x^{\prime \prime}+f(x) x^{\prime}+h(x)=0$ where $f(x)>0$ and $x h(x)>0$ for $x \neq 0$ and such that $f, h$ are continuous. b) Additionally, show that if $$ \lim _{|x| \rightarrow \...
Ri-Li's user avatar
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0 answers
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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-...
J Peterson's user avatar
3 votes
1 answer
98 views

Change of basis in an Hamilton system

I came across the System of equations $\begin{equation} \left\{ \begin{aligned} \frac{d}{dt}p &= - \omega^2q \\ \frac{d}{dt}q &= p \\ p(0)&=p_0,q(0)=q_0 \end{aligned} \right. \end{...
The Lion King's user avatar
1 vote
0 answers
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Singularity in Hamiltonian for a spherical pendulum

I am performing a numerical simulation of the example in this page, but am having problems because the Hamiltonian (specifically the $\phi$ momentum part of the kinetic energy) is undefined (infinite?)...
m4r35n357's user avatar
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1 answer
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Constructing ODEs from a Hamiltonian

I'm reading this book and came across this specific Hamiltonian in equation (5.2). My question is how to construct the corresponding system of ODEs explicitly. I am familiar with doing so in a simple ...
Rudinberry's user avatar
2 votes
0 answers
56 views

Are classical many-body systems ergodic?

We take a general Hamiltonian system $$ H(\boldsymbol x_1,\cdots,\boldsymbol x_n,\boldsymbol v_1,\cdots,\boldsymbol v_n) = \frac{1}{2}\sum_{i=1}^nm|\boldsymbol v_i|^2+\sum_{1\leq i < j \leq n}U(|\...
Mr. Egg's user avatar
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3 votes
1 answer
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Question about the period of specific Hamiltonian flows

Consider the Hamiltonian of the form $H(q,p)=p^2/2 + |q|^{\beta}/\beta$ for $\beta\in (1,2)$. In the case of $\beta=2$, this is simply the harmonic system and we know that all contours, i.e. those of ...
nomadicmathematician's user avatar
0 votes
1 answer
29 views

Are my answers correct for this Lagrange + Hamilton Excess function?

I have been given the following Question: Consider the linear Lagrangian function $L$ in $\mathbb{R}^2$ given by $$ L(t,x,\dot x)=\alpha(t,x)+\beta(t,x)\dot x, $$ and the corresponding variational ...
Martin Sieburg's user avatar
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Calculation of the Lie derivative of the fundamental one-form in three different ways

I am a physicist who is trying to understand more formal differential geometry in the context of classical mechanics. I came across three ways of computing the Lie derivative of differential one-forms....
Ben's user avatar
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Relating the Lagrangian and Hamiltonian dynamics

Consider $M$ to be a compact smooth manifold , and $L_t$ a Lagrangian on $TM$ that satisfies $\partial_{vv}L\geq l_0 I$. Then we are able to define a hamiltonian $H_t:T^*M\rightarrow \mathbb{R}$ by ...
Someone's user avatar
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1 vote
0 answers
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How to derive directly from Hamilton equations to the Euler-Lagrange equations?

We have the function (Lagrangian):$$L(\theta, \phi, \dot{\theta}, \dot\phi)=\frac{mb^2}{2}(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)+mbg\cos(\theta)$$ And I have found the Hamilton function $H$ and ...
Lifeni's user avatar
  • 558
4 votes
1 answer
360 views

Proving that implicit midpoint method for Hamiltonian systems is symplectic using a criterion

The implicit midpoint rule is defined as $$y_{n+1}=y_n+hJ^{-1}\nabla H\left(\frac{y_{n+1}+y_n}{2}\right).$$ where $y=(p,q)$. I know how to prove that this method is symplectic by hand, using the ...
Jiu's user avatar
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7 votes
0 answers
180 views

modified Hamiltonians for symplectic methods

I'm interested in methods for numerically integrating Hamiltonian systems $$\begin{align} \dot q & = +\frac{\partial H}{\partial p} \\ \dot p & = -\frac{\partial H}{\partial q} \end{align}$$ ...
Daniel Shapero's user avatar
0 votes
1 answer
191 views

How to use LaSalle principle for this Lyapunov (Hamiltonian)?

I am analyzing this problem, and some questions has appeared to me. 1- In case 2 and 3, which Hamiltonian should I choose? 2- I did not understand case 4? Could you enlighten me please. Let $\alpha$ ...
Mr. Proof's user avatar
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1 vote
0 answers
35 views

Numerical integration technique for equations with a streamfunction?

In fluid dynamics, it is common to encounter velocity fields that can be written in terms of a streamfunction. In two dimensions, $$ \dot x = \dfrac{\partial{\psi(x, y)}}{\partial{y}}, \;\;\;\dot y = -...
wil3's user avatar
  • 137
3 votes
1 answer
487 views

Composition of flow maps of Hamiltonian systems

Given a pair of autonomous Hamiltonian vector fields $X_H,X_K\in\mathfrak{X}(M)$, with flow maps which are respectively $\Phi^t,\Psi^s$, is $\Phi^t\circ \Psi^s$ the $(t+s)-$flow map of some ...
Dadeslam's user avatar
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1 vote
1 answer
124 views

Small oscillations of a dynamical system near stable equilibrium points

I'm having problems solving this lagrangian dynamical system: Let $P$ and $Q$ be two points in $\mathbb R^2$ s.t. $P\in\Gamma_1\equiv y=x^2$ and $Q\in\Gamma_2\equiv y=-x^2$. The two points are ...
Vajra's user avatar
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1 vote
0 answers
75 views

Does every Hamiltonian equation can be written in a some system of coordinates as a canonical Hamiltonian system

Does every Hamiltonian equation can be written in a some system of coordinates as a canonical Hamiltonian system i.e. $\exists(q,p)$ such that $$\begin{equation*} \begin{cases}\dfrac{dq}{dt}=\...
Niser's user avatar
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