Questions tagged [hamiltonian-mechanics]
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11 questions
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An algebraic Hamiltonian vector field with a finite number of periodic orbits (2)
Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
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An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)
Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question.
Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ(...
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are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?
I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
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Hamiltonian, Lagrangian and Newton formalism of mechanics
If my thinking is wrong please let me know. I have little knowledge on beyond-college physics.
For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,...
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Generalizing "variation of parameters"
I'm stuck on generalizing an ODE formula and could use your help!
One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here $z(t)\in\mathbb{R}^n$...
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When does a Lagrangian dynamical system have an equivalent Hamiltonian description?
Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$
(i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q_{i},\dot{q}_{i},t)$, $i=1,2,......
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Practical example of Hamiltonian reduction
I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p_1, \...
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Periodic orbit for certain Hamiltonian on the tangent bundle
In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point.
Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...
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Global reduction of Hamiltonian with an integral of motion (Poincare' reduction)
This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question.
Background
I refer to the following concepts:
Liouville ...
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How does the symplectic form $\omega$ manifests itself in the Euler-Lagrange equation? + Extreme confusion with time
Let $\omega$ be a symplectic manifold on $\mathbb{R}^n$ and the smooth function $H : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a Hamiltonian. For $p,q \in \mathbb{R}^n$ let us assume that
\...
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How to express the Euler-Lagrange equation in arbitrary coordinates where $\omega \neq \sum dp_i \wedge dq_i$
I posted my questions in a previous post MO, but it seems that a more refined version for question on the Euler-Lagrange equation is needed. So, I post my question again.
In standard symplectic ...
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