All Questions
Tagged with celestial-mechanics classical-mechanics
29
questions
15
votes
2
answers
512
views
Regularization: What is so special about the Coulomb/Newtonian and harmonic potential?
I wanted to know if the procedure for regularization of the Coulomb potential outlined in Celletti (2003): Basics of regularization theory could be generalized to arbitrary polynomial potentials. So ...
- 9,890
12
votes
3
answers
893
views
What could cause an asymmetric orbit in a symmetric potential?
My question can be summarized as:
Given a potential with a symmetry (e.g. $z\rightarrow-z$), should I expect orbits in that potential to exhibit the same symmetry? Below is the full motivation for ...
- 18.5k
5
votes
1
answer
191
views
How are Lie series used as canonical transformations in perturbation theory?
I have a few questions on how to use Lie series as a canonical transformation, which are widely used in perturbation theory (celestial mechanics).
I know that these series are related to a Taylor ...
- 331
4
votes
0
answers
129
views
Why Kepler problem is equivalent to a free particle on 4 dimensional sphere?
In trying to understand Laplace-Runge-Lenz vector, I read in Wikipedia
that the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four dimensional hypersphere....
- 449
3
votes
2
answers
897
views
What is the "associated scalar equation" of equations of motion?
In an essay I am reading on celestial mechanics the equations of motion for a 2 body problem is given as:
$$\mathbf{r}''=\nabla(\frac{\mu}{r})=-\frac{\mu \mathbf{r}}{r^3}$$
Fine. Then it says the "...
- 1,517
3
votes
1
answer
420
views
Feynman's Lost Lecture: what is the significance of $\frac{d\mathfrak{v}}{d\theta}=-\frac{GMm}{\left|\mathfrak{L}\right|}\hat{\mathfrak{r}}?$
My question pertains to a fact used by Richard Feynman in his so-called
Lost Lecture. http://books.wwnorton.com/books/Feynmans-Lost-Lecture/.
I have only skimmed the book, so I have much more to learn ...
- 2,015
3
votes
0
answers
43
views
Explicit construction of action-angle variables for the two-fixed-centers problem
After studying action-angle variables and Eulers two-fixed-center problem in a course on mechanics and symplectic geometry, I understand that a two-fixed-center system is Liouville integrable and ...
3
votes
1
answer
429
views
Why do the orbit equations have to be symmetric about two axes even the orbit is not bounded?
In the book of Classical Mechanics by Goldstein, at page 88, it is given that:
$$
\frac{d^{2} u}{d t^{2}}+u=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right) .
$$
The preceding equation is such ...
- 2,283
3
votes
0
answers
447
views
Story about a mathematician, a dinner party, and the three-body problem
I remember dimly hearing a story, coincidentally also at a dinner party, and I was trying recently to track the details down with no success. I was hoping someone here might have also heard this story ...
- 139
2
votes
2
answers
165
views
A doubt in a Wikipedia article discussing Bertrand's theorem
Wikipedia while deriving Bertrands theorem writes after some steps:
...For the orbits to be closed, $β$ must be a rational number. What's more, it must be the same rational number for all radii, ...
- 1,270
2
votes
1
answer
126
views
A doubt in a Wikipedia article discussing Bertrand's Theorem in Central force motion
Wikipedia on Bertrand's theorem, when discussing the deviations from a circular orbit says:
...The next step is to consider the equation for $u$ under small perturbations ${\displaystyle \eta \equiv ...
- 1,270
2
votes
2
answers
231
views
Is there a variable mass Verlete like integration algorithm?
I'm currently modeling the explosion of a star. For my simulation, I use a Verlete like integration algorithm. This is quite common in celestial mechanics modeling. The thing is that now that I have ...
2
votes
1
answer
1k
views
Angular momentum components as independent integrals of motion
I was told that in order to solve the Kepler problem (6 degrees of freedom in total) you have to proceed, step by step, to reduce those degrees of freedom using the integrals of motion. You do so ...
- 65
2
votes
0
answers
151
views
Harmonic and subharmonic orbits in central fields
Using Newton's theorem of revolving orbits one can easily obtain orbits for central forces containing inverse cube terms, such as
$$F(r)=F_0(r)+\frac{(1-k^2)|B|}{r^3},$$
from known orbits for $F_0$. ...
- 17.8k
1
vote
3
answers
1k
views
Does it take energy (in joules) to keep the moon in orbit around the Earth?
If energy is force times distance and I use the Newtonian formula to calculate the force between the earth and the moon
$$ F=\frac{G m_1 m_2}{r^2} , $$
then multiply it by the circumference of the ...
- 163