70
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) ...
- 178k
39
votes
Accepted
Euler's Master's Thesis
Martin Mattmüller, in his article Leonhard Euler, seine Heimatstadt und ihre
Universität on Euler's hometown Basel, writes that this public talk (not a dissertation or written thesis), which Euler ...
- 32.2k
20
votes
Accepted
What do physicists mean by a topological quantum gravity theory
Physicists here. The input for a physical theory is always some topological space and some structure (such as a metric) that depends on the specific context. The dynamics are invariant under the ...
20
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: ...
- 29.5k
17
votes
Decidability of 3 body problem
In Church's thesis meets the N-body problem Warren Smith argues that unsimulable physical systems exist in Newton’s laws of gravity and motion for point masses, because an uncountably infinite number ...
- 178k
14
votes
Accepted
Decidability of 3 body problem
The paper Undecidability in $\mathbb{R}^n$: Riddled Basins,
the KAM Tori, and the Stability of the Solar System by Matthew W. Parker (Philosophy of Science 70 (April 2003), 359–382) comes close to ...
- 78.3k
8
votes
Euler's Master's Thesis
As Franz says, it is impossible to know for definite which view the young Euler supported.
However, Newton had already disposed of the Cartesian vortex theory in the Principia which was published ...
- 4,958
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 ...
- 5,472
5
votes
Accepted
Inverse square-law as a positive definite kernel?
Please look pages 141-147 of my book:
S. Saitoh and Y. Sawano, Theory of Reproducing Kernels and Applications,
Developments in Mathematics 44, Springer (2016).
2023.5.4.20:36 https://doi.org/10.1007/...
5
votes
Electromagnetic energy in Lovelock gravities
The coupling of electromagnetism (including Born-Infeld nonlinearities) to Lovelock gravity has been studied in Magnetic Branes in Third Order Lovelock-Born-Infeld Gravity. The nonlinearities in the ...
- 178k
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 ...
- 2,271
3
votes
What are the applications of spin geometry?
I understand the question as requesting applications of the geometry of spin manifolds outside of pure mathematics. There are many, starting with the Dirac equation for the spinor of a relativistic ...
1
vote
Accepted
Mach's principle, Newton's law and Hilbert sphere?
I will try to answer my own question, and allow me to change a little bit the notation and meaning, since it was not so difficult as first thought, it was.
First let us look at the situation where two ...
- 2,462
1
vote
Inverse square-law as a positive definite kernel?
The answer to this question, can be given a meaning with the Schoenberg criterion from which the following explanation is borrowed:
The function $\Psi \colon X \times X \to \mathbb{R}$ is said to be a ...
- 2,462
1
vote
Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?
There are some amazing aspects of hidden symmetry - Bertrand’s theorem connection. The first surprise is that it seems Runge-Lenz vector has relativistic (!) origin: https://www.sciencedirect.com/...
- 16.4k
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