All Questions
Tagged with celestial-mechanics classical-mechanics
29
questions
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 ...
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
0
votes
0
answers
113
views
Applications of Hamiltonian formalism in classical or celestial mechanics
I am looking for a reference (or just a brief explanation) to applications of the Hamiltonian formalism to classical mechanics, e.g. to planetary motion.
In all known to me textbooks on classical ...
0
votes
2
answers
388
views
DART crash on Dimorphos: computation of orbital period change
What is the distribution of expected changes in the period of Dimorphos' orbit around Didymos when the spacevehicle DART crashes against it?
- 1,106
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
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
1
vote
1
answer
203
views
Bertrands theorem, Hooke's law and closed orbit [closed]
Bertrand's Theorem says: the only forces whose bounded orbits imply closed orbits are the Hooke's law and the attractive inverse square force.
I'm looking at the Hooke's law $f=-k r$ and try to see ...
- 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 ...
0
votes
1
answer
269
views
Derivation of the equation of a hyperbolic orbit from the conic section expression derived via the orbit equation
So I'm looking to derive the equation of a hyperbolic orbit from the general expression for a conic section $$r=\frac{l}{e\cos\theta+1}$$ that you get out of solving the orbit equation for an inverse-...
1
vote
0
answers
69
views
Time taken to collide [duplicate]
Two point masses m1 and m2, separated initially by distance d, move towards each other under mutual gravitational force.
Find the time they take to collide?
The main problem I'm having is to solve the ...
1
vote
0
answers
49
views
How did most of the math and physics formulae that govern our lifestyle and help us in space exploration come into being? [closed]
I realise that this isn't a very academic question, but after watching movies like First Man and Interstellar, it got me wondering:
How did all these formulas that we use on a regular basis come into ...
- 131
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
1
vote
0
answers
168
views
Planar Precession Frequency of Orbit
What is the general relation between orbital precession $\Phi$, orbital frequency $\Omega$ and a radial perturbation frequency $\omega$?
For certain cases the answer is "clear", for example:
1) If $\...
- 683
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
-2
votes
1
answer
115
views
Spirals in newtonian celestial mechanics?
I know Kepler's laws, Newton's laws, and that conic sections are the trajectories of noncolliding two point masses. But I wonder about a point mass A eventually colliding with point mass B.
In ...
- 926
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
1
vote
0
answers
53
views
Sign of the action for the harmonic osccilator?
I am confused about the derivation of the action $S(x,\mathbf{J})$ for a harmonic oscillator as given at page 219 in "Galactic Dynamics", J.Binney-S. Tremaine, 2nd Ed. 2008. The part of the derivation ...
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
-2
votes
1
answer
223
views
Impact of Moon's gravitational pull on Earth [closed]
What speed does the Moon's gravitational pull impart to Earth?
0
votes
1
answer
247
views
Problem: Spectroscopy of a binary system
The Problem is:
For a binary system (2 Stars) with Orbital Period of $P =4.822 days = 416620.8 second$ and inclination $i=90$ and with speeds very less than $3 .10^8 m/s$. Their orbital planes around ...
- 115
0
votes
0
answers
37
views
How can I measure the stability of a many body gravitational system?
Suppose I have an N body planetary system interacting via gravity. Suppose I know the positions and momenta at t=0. How do I know if this system is stable (indefinitely)? By stable I mean the ...
- 1,601
1
vote
1
answer
165
views
Stability of the classical helium atom
Let us forget about quantum mechanics and confine ourselves to classical mechanics.
The Hamiltonian for a classical helium atom would be
$$ H = \frac{p_1^2 + p_2^2}{2m } - \frac{Z}{r_1} - \frac{Z}{...
- 4,122
3
votes
2
answers
897
views
What is the "associated scalar equation" of equations of motion?
In an essay I am reading on celestial mechanics the equations of motion for a 2 body problem is given as:
$$\mathbf{r}''=\nabla(\frac{\mu}{r})=-\frac{\mu \mathbf{r}}{r^3}$$
Fine. Then it says the "...
- 1,517
0
votes
1
answer
51
views
What does an $n$-body system with constant $T$ and $U$ look like?
Can someone give an example of a system where the kinetic $T$ and potential $U$ energy are constant (but not zero)?
Here's what I have in mind: Say you have $n-1$ satellites of negligible mass ...
- 1,517
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
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
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