# Tag Info

### 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) ...
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### 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 ...
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### 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: ...
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### 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 ...
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 ...
• 155k
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, ...
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### 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 ...
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### 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 ...
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 $M$.
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 (...
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• 5,690