The classical-mechanics tag has no wiki summary.

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### Herpolhode equation

Poinsotâ€™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 ...

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### Minimal surface which divides a convex body into two regions of equal volume

Question. Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume?
Background/motivation.
A 2D version of ...

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### 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 ...

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894 views

### Dense orbits in billiards

This should be true in a more general setting, but for simplicity consider billiards that are connected, compact subsets of the plane with boundary $C^2$ except at finitely many points. A ball (or a ...

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**1**answer

967 views

### On the non-rigorous calculations of the trajectories in the moon landings

In a paragraph written by a person emphasizing that rigour is not everything in mathematics (I wish I had written down the details), it was stated that the moon landings would have been impossible ...

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### Bertrand theorem - central forces

Here is a version of Bertrand theorem. Let us consider a force $F(r)$ which depends only on the distance to a given point. If all trajectories which remain bounded are closed, then either $F(r)=ar$ ...

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### Floating polyhedra with fair equilibria

Is there a homogeneous convex polyhedron
which floats so that some subset (perhaps all) of its faces
is distinguished as "up" (above the water line)
in stable equilibrium, each face with ...

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**1**answer

558 views

### Which motion is exclusive in 3D or higher dimensions?

Hi guys,
I have a simple question
Linear movement can be found in 1D, 2D and 3D world objects
Rotation can be found in 2D and 3D world objects.
Now, are there any kind of motion can only be found ...

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### turbulence as an unsolved problem of classical mechanics

Why is it that turbulence is considered to be an unsolved problem of classical mechanics? What is meant by "unsolved"? Don't the Navier-Stokes equations apply to turbulent flows? It's difficult to ...

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### Fair but irregular polyhedral dice

I am interested in determining a collection of geometric conditions that will guarantee that a convex polyhedron
of $n$ faces is a fair die in the sense that, upon random rolling, it has an equal ...

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### Poincare Recurrence and Dense Sets

This is kind of a spin-off of the question asked here. Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The ...

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### How quickly will billiard trajectories cluster?

Suppose you launch $n$ point-particles on
distinct reflecting nonperiodic billiard trajectories
inside a convex polygon. Assume they all have the same speed.
Define an $\epsilon$-cluster as a ...

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**1**answer

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### What are the canonical and earliest references to trivial symmetries in gauge systems?

I am trying to find canonical references and the history of trivial symmetries.
The earliest text book reference I can find is on page 69 of Quantization of Gauge Systems by Henneaux and Teitelboim.
...

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2k views

### $\exists$ a shot in ideal pocket billiards?

Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with
the game idealized in that no spin is placed on the cue ball in
the initial shot, all collisions between billiard ...

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**1**answer

1k views

### Hanging a ball with string

What is the shortest length of string that suffices to hang
a unit-radius ball $B$?
This question is related to an earlier MO question, but I think different.
Assume that the ball is ...

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2k 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, ...

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### Is symplectic reduction interesting from a physical point of view?

Do you think that symplectic reduction (Marsden Weinstein reduction) is interesting from a physical point of view? If so, why? Does it give you some new physical insights?
There are some possible ...

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### Two interacting bodies in an external field

Hope, MO is the right place for this question (if not so: where would you pose it?).
Consider a two-body system in classical mechanics. As long as the interaction depends only on the distance of the ...

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### 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 ...

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**0**answers

401 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 ...

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**1**answer

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### Polygonal billards programs

I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure.
It was a good exercise, but at this point I wonder if ...

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3k views

### Angle Maximizing the Distance of a Projectile

It is well-known that to maximize the horizontal distance traveled by a projectile fired from the ground at a given speed, one should fire it at a $45^\circ$ angle. What's less-known, though not too ...

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**1**answer

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### Rolling a convex body: Geodesics vs. rolling curves

What are the curves of contact on a convex body $B$ rolling down an inclined plane?
Assume $B$ is smooth, and there is sufficient friction to prevent slippage.
Certainly, one can develop a geodesic ...

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### On linear independence of exponentials

Problem.
Let $\{\lambda_n\}_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is ...

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### Parabolic envelope of fireworks

The envelope of parabolic trajectories from a common launch point
is itself a parabola.
In the U.S. this weekend many will have a chance to
observe this fact direcly, as the 4th of July is ...

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### Functional Analysis and its relation to mechanics

Hi I'm currently learning Hamiltonian and Lagrangian Mechanics (which I think also encompasses the calculus of variations) and I've also grown interested in functional analysis. I'm wondering if there ...

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### Generalizing square wheels rolling on inverted catenaries

It is not uncommon to see in a science museum a bicycle with
square wheels that rides smoothly over a washboard-like
surface made from inverted catenary curves (e.g., at the Münich museum).
The ...

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**4**answers

887 views

### When sticks fall, will they weave?

Imagine $n$ $z$-vertical sticks uniformly spaced around a unit-radius circle in the $xy$-plane.
At $t{=}0$, each is randomly $\epsilon$-perturbed from the vertical, and they fall under
the influence ...

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### Catenary curve under non-uniform gravitational field

The catenary curve is the shape of a chain hanging between two equal-height poles under the influence of gravity. But the derivation of the (hyperbolic cosine) curve equation from the physics ...

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### Classical mechanics motivation for poisson manifolds?

Suppose I want to understand classical mechanics.
Why should I be interested in arbitrary poisson manifolds and not just in symplectic ones?
What are examples of systems best described by non ...

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### How to calculate the rolling resistance of a wheel over an obstacle? [closed]

Imagine a bicycle travelling at speed, and then rolling over a log. What are the principles behind calculating the force that is required to roll a wheel over an obstacle?

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### Fastest Rolling Shape?

The following questions occurred to me.
This is not research mathematics, just idle curiosity.
Apologies if it is inappropriate.
Suppose you have a fixed volume V of maleable material,
perhaps ...

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### How can I conclude that I live in a solar system?

Well, this is an awkward question and I don't know if it is mathematical enough for MO (I'm sorry if not) but I'll try it: What observations in the coordinate system centered in my fixed position on ...

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### 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 ...

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### 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 ...

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### Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector

Consider an idealized classical particle confined to a two-dimensional surface that is frictionless. The particle's initial position on the surface is randomly selected, a nonzero velocity vector is ...