66 votes

Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?

The gravitational or Coulomb potential has a "hidden" symmetry (hidden in the sense that it does not follow from the rotational symmetry). The resulting integral of the motion (the Runge-Lenz vector) ...
21 votes

Hamiltonian, Lagrangian and Newton formalism of mechanics

Each of the different formalism of classical mechanics has its advantages and disadvantages. However, in the end all three frameworks tend to be equivalent, and thus the following list is very ...
  • 5,192
19 votes

Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?

There are: Bertrand’s theorem, which says that the isotropic oscillator and Kepler potentials are the only analytic radial ones all of whose nonrectilinear bounded orbits are closed. (Recommendation: ...
18 votes

Hamiltonian, Lagrangian and Newton formalism of mechanics

The three formalisms of classical mechanics, i.e. the Newtonian, the Lagrangian (analytical mechanics) and the Hamiltonian (canonical formalism) are generally not equivalent to each other -at least ...
11 votes
Accepted

Poincaré recurrence and its implications for statistical physics and the arrow of time

Since the question is about physical implications of Poincaré recurrence one should take both quantum effects and gravitational effects into consideration. Quantum mechanics does not spoil the ...
10 votes
Accepted

Why is every Hamiltonian system locally integrable?

For non-singular Hamiltonian systems, you can see it as a consequence of a slight generalization of the Darboux Theorem which is known as the Carathéodory--Jacobi--Lie Theorem (see, e.g. Libermann, ...
  • 4,196
8 votes
Accepted

Momentum a cotangent vector

The Lagrangian is a function on the tangent bundle $L:TM\rightarrow\mathbb{R}$. Given a point $q\in M$ and a Lagrangian, we can define a function $L_q:T_qM\rightarrow \mathbb{R}$ using the simple ...
8 votes
Accepted

Hamilton equations for Classical Field Theory

There is a fundamental misunderstanding in your translation of Hamilton's formalism to classical field theory, which pertains to the proper identification of dynamical variables. In classical ...
7 votes
Accepted

Symplectic reduction of 4-manifolds with circle actions

By a result of Hui Li https://arxiv.org/abs/math/0605133, the fundamental groups of $M_{red}$ and $M$ are isomorphic, so one can deduce the genus of $M_{red}$ from the fundamental group of $M$.
  • 6,558
6 votes

Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?

Here is an interpretation using symmetry reduction, but without explicitly using the Lenz-Runge vector (it's essentially an extended version of the example given in Cushman & Bates "Global aspects ...
  • 5,192
6 votes

Applications of Hamiltonian formalism to classical mechanics

The Poincaré-von Zeipel method in celestial mechanics relies on canonical transformations of the Hamiltonian to separate fast and slow degrees of freedom in a solar system. See, for example, A note on ...
6 votes

Poincaré recurrence and its implications for statistical physics and the arrow of time

The Poincare recurrence (or, more general, the ergodic theorem that says that a system will, over time, evolve through essentially all microscopic states that are consistent with the total energy, ...
  • 862
5 votes
Accepted

Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus

1) is correct: by saying “hamiltonian diffeomorphism of” you imply that the torus has a symplectic structure $\omega$ and the diffeo is (something like) time 1 flow of a hamiltonian vector field $X$: $...
5 votes
Accepted

Reference Request: KAM Theory

There are many good books. I can recommend two: S. Sternberg, Celestial mechanics, Part 2, W. A. Benjamin Inc., NY 1969 V. I. Arnold, Geometrical methods in the theory of ordinary differential ...
5 votes

Constants of motion for Droop equation

A complete global analysis of the Droop equations is carried out in Lange, Kenneth; Oyarzun, Francisco J., The attractiveness of the Droop equations, Math. Biosci. 111, No. 2, 261-278 (1992). ...
5 votes
Accepted

Exact solution to a periodic linear ODE sought

Rather incredibly, your (corrected) system does have a closed-form solution, which I found with Maple's help. $$ x(t) = 1+4\,\cos \left( 2\,t \right) +3\,\sqrt {8\,\cos \left( 2\,t \right) + 17}$$ $y(...
4 votes

Non-Hamiltonian actions in physics

The master thesis Nonholonomic Dynamical Systems by Brett Ryland contains several examples of non-Hamiltonian systems from classical physics: the dynamics of a laser and the evolution of a gas flame (...
4 votes
Accepted

When does a Lagrangian dynamical system have an equivalent Hamiltonian description?

Here's what I have done: $\bullet$ Let the Lagrangian $L(q_{i},\dot{q}_{i},t)$, which under the point transformations $$ \{q_{i}\}\leftrightsquigarrow\{Q_{i}\} $$ given by the invertible relations $...
4 votes

Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$

There is a wonderful trick - I think promulgated by Moser - for viewing Hamiltonian flows on the cotangent bundle of the sphere as the reduction of a Hamiltonian flow on the ambient phase space of (...
4 votes
Accepted

What are the compact Lagrangian submanifolds of a twisted cotangent bundle?

Let's begin by pointing out the following: you will not find monotone examples for the simple reason that a nontrivial such deformations creates a class of nonzero symplectic area, while the Chern ...
  • 546
4 votes

Hamiltonian, Lagrangian and Newton formalism of mechanics

In response to (1), a key advantage of Hamilton's equations of motion is that they remain invariant under a large class of "canonical" transformations, $(x,p)\mapsto (Q(x,p),P(x,p))$ for some scalar ...
4 votes

Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?

The action-angle variables of the two-body graviational problem ('Kepler problem') are widely used in celestial mechanics community. These are called 'Delaunay variables' and make the toric structure ...
4 votes

Applications of Hamiltonian formalism to classical mechanics

E.T. Whittaker's A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge, multiple editions from 1904 onwards) provides applications, particularly to celestial mechanics. R. ...
3 votes
Accepted

Points with finite stabilizer in Hamiltonian torus actions

If I understand the question correctly, what you are after is an effective Hamiltonian action of $\mathbb{T}^m$ on a (closed) symplectic manifold $(M,\omega)$ such that there exists a point $p \in M$ ...
3 votes

Applications of Hamiltonian formalism to classical mechanics

Historical References: "A History of Mechanics" by Rene Dugas is highly recommended reference, which includes discussion of Hamilton's work and its development. Below I will summarize some ...
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3 votes

Mechanical systems with their configuration space being a Lie group

Many of the standard classical mechanics examples have continuous symmetries. Most are invariant under translations in time. Many are invariant under spatial translations and rotations, e.g., isolated ...
3 votes
Accepted

Uniform continuity of Hamiltonian flow

Denote by $L$ the Lipschitz constant of the Hamiltonian vector field $\mathfrak{X}$ and by $\varphi_t$ the flow generated by $\mathfrak{X}$. Then for any $x, y \in \mathbb{R}^{2n}$, by the chain ...
2 votes
Accepted

Generalizing HJB equation for a terminal stopping time

At least formally, the given HJB equation with possibly some additional Dirichlet boundary conditions (more on this point below) does hold for the value function associated to the stopped or absorbed ...

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