All Questions
10 questions
4
votes
0
answers
235
views
Generalising Bäcklund transform to solve $\omega''(t)=t\sin\omega(t)$
Bäcklund transformations may be used also in ODE to solve non-linear problems; for instance, it's well known that for the equation
$$
\frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=\sin\omega
\tag{*}\label{...
- 134
15
votes
9
answers
4k
views
Newton equations, second order equation and (im)possible motions
I am am currently studying Newtonian mechanics from a conceptional and axiomatic point of view. Now, if I am not mistaken, one (but surely not all) statement of Newtons second law about nature is, ...
- 1,222
13
votes
3
answers
633
views
Random N-body problem
Suppose there are $N$ unit-mass particles whose initial positions
are uniformly distributed in a unit-radius disk.
Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...
- 151k
4
votes
2
answers
592
views
Can one obtain this ODE as an Euler-Lagrange equation?
Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...
- 53
17
votes
5
answers
2k
views
2- and 3-body problems when gravity is not inverse-square
Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing
as $1/d^p$ for distance separation $d$ and some power $p$.
Two questions:
Presumably the 2-body ...
- 151k
101
votes
1
answer
8k
views
Dropping three bodies
Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...
- 8,336
0
votes
1
answer
89
views
underdamped oscillation with quadratic decay
I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form:
...
- 169
4
votes
0
answers
116
views
Dynamics of pairwise distances in the $n$-body problem
Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...
3
votes
2
answers
947
views
Herpolhode equation
Poinsot’s construction describes the motion of a freely rotating rigid body in terms of an ellipsoid rolling on a plane. (http://www.phys.ttu.edu/~huang24/Teaching/Phys5306/CH5C.pdf), and the path of ...
- 133
6
votes
3
answers
450
views
Do there exist small neighborhoods in a classical mechanical system without pairs of focal points?
The question I will ask makes sense in much more generality, but I will leave the translation to the experts, since I'm only looking for a special case (and it would not surprise me if the answer does ...
- 54.6k