The classical-mechanics tag has no usage guidance.

**86**

votes

**0**answers

6k 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 ...

**11**

votes

**0**answers

428 views

### Functions approximated by rolling epicycle curves

Imagine a decreasing sequence of (positive) radii $r_1 > r_2 > r_3 > \cdots$
and a series of nested circles $C_1 \supset C_2 \supset C_3 \supset \cdots$
with these radii,
initially each ...

**9**

votes

**0**answers

280 views

### Periodic orbits of a spinning ball in a square

Periodic orbits of a billiard ball bouncing in a square have been well-studied.
I am seeking similar analysis of what is sometimes called a rough ball, one
whose high friction causes it to pick up ...

**8**

votes

**0**answers

202 views

### Billiards with incompatible regions

An existing question asks whether "almost every" two-dimensional billiard possesses at least one orbit that is dense in its interior. My question is about the following set of strong ...

**7**

votes

**0**answers

274 views

### Question about theorem in Arnold's book on action-angles variables

I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question)
If you don't have the book or ...

**7**

votes

**0**answers

101 views

### Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes.
(An ...

**6**

votes

**0**answers

419 views

### Differential equation of line tangent to caustics

This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most ...

**5**

votes

**0**answers

550 views

### Egg-ovoid rolling down an inclined plane

I am seeking a mathematical analysis of an egg-ovoid rolling down an inclined plane,
for pedagogical reasons.
It is well-known folk lore that the shape of an egg prevents it from rolling away from
...

**5**

votes

**0**answers

200 views

### Generalization of the non-existence of a monostatic planar body

Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only
one orientation of stable equilibrium and one orientation of unstable ...

**3**

votes

**0**answers

145 views

### Dynamics of electrons on a sphere

Suppose one place $n$ electrons closely surrounding the north pole of a sphere, forming
a perfect planar regular $n$-gon:
Q1.
What will happen if the ...

**3**

votes

**0**answers

86 views

### A taut string of equilateral triangles

Let $T$ be a unit edge-length equilateral triangle composed of three cylinders each
of (small) radius $r>0$. (By "small" I mean approximately $< 0.1$.)
Think of $T$ as a physical, rigid ...

**3**

votes

**0**answers

148 views

### What happens when Appell-Chetaev's rule for constrained mechanical systems is not applicable?

Background:
Let be given a mechanical system whose configuration space is a manifold $Q$, and the kinetic energy is a metric $K$ on $Q$, in presence of a potential function $V$.
Let us identify the ...

**3**

votes

**0**answers

531 views

### Find a second integral for Arnold's example

Consider Arnold's example for Arnold diffusion 1964.
$$H=I_1^2/2+I_2^2/2+\epsilon(1-\cos\theta_2)(1+\mu(\sin\theta_1+\sin t)) $$
We can first make it a system of three degrees of freedom.
Then we ...

**2**

votes

**0**answers

115 views

### Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom

Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation
$$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$
can ...

**2**

votes

**0**answers

116 views

### How to find solutions of non-linear ODE with particular BCs

What are some methods, numerical or otherwise, of finding solutions to nonlinear ODEs that satisfy particular boundary conditions? In particular, I'm looking for curves y(s) constrained to a ...

**2**

votes

**0**answers

230 views

### In search for a more geometric proof of a result of van der Schaft and Maschke on nonholonomic mechanics.

Edit: Now I have found something that appears to answer my own question. It is section 2 in the paper "On Submanifolds and Quotients of Poisson and Jacobi Manifolds" by Ch.-M. Marle. (There, he ...

**1**

vote

**0**answers

47 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.
...

**1**

vote

**0**answers

136 views

### Rigid-body in a central field: orbital and attitude motion

Question
I would like to find a nice set of explicit coordinates for the family (parametrised by angular momentum) of reduced systems representing a rigid-body in a central field
in which the orbital ...

**1**

vote

**0**answers

265 views

### Dissipative Hamiltonian System with a Periodic Force

Let $H:P \to \mathbb{R}$ be a Hamiltonian on a symplectic manifold $(\omega,P)$ and let $X_H: P \to TP$ be the Hamiltonian vector-field. Let $F:P \to T^*P$ be a dissipative force field such that for ...

**0**

votes

**0**answers

173 views

### Transformation of the dynamics of mechanical system under coordinate change

It is well known that the dynamics equation for a mechanical system (e.g. a robotics manipulator) is given be the Euler-Lagrange equation which takes the particular form (in the simplified version),
...