Questions tagged [classical-mechanics]
Mathematics of classical mechanics, including Hamiltonian mechanics, Lagrangian mechanics, applications of symplectic geometry to mechanics, deterministic chaos, resonance etc.
191 questions
32
votes
2
answers
2k
views
Gently falling functions
I wonder if it is possible to characterize the class of
gently falling functions, which I would like to define
as follows.
Let $g(x)$ be a $C^2$ function defined on an interval
$R \subseteq \mathbb{R}$...
- 151k
8
votes
1
answer
2k
views
Calculating the geodesic equation for a particular set of phase-space coordinates
Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ ...
- 8,532
24
votes
2
answers
1k
views
Billiard dynamics for multiple balls
I am interested to learn to what extent results on billiards
in polygons have been extended to multiple balls.
Assume the balls have equal radii and the same mass,
the same initial speed, and all
...
- 151k
21
votes
1
answer
1k
views
Which convex bodies roll along closed geodesics?
An ellipsoid could be rolled (without slippage) on a horizontal plane so that its point
of contact traces out a closed geodesic on its surface:
...
- 151k
10
votes
1
answer
2k
views
Equations for an algebraic gömböc
A gömböc is a $3$-dimensional convex body (having uniform density) which has exactly one stable and one instable equilbrium position (see https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c).
Such a convex ...
- 17.9k
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
2
answers
656
views
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 the ...
- 22.3k
8
votes
0
answers
246
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 counter-examples:...
- 131
5
votes
3
answers
2k
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 ...
- 445
14
votes
1
answer
2k
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 ...
- 4,351
8
votes
3
answers
5k
views
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 $F(r)=ar$ ...
- 551
11
votes
2
answers
1k
views
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 equal ...
- 151k
3
votes
1
answer
697
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 ...
- 101
5
votes
3
answers
3k
views
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 ...
- 51
63
votes
8
answers
14k
views
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 $1/n$...
- 151k
5
votes
2
answers
996
views
Poincaré 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 ...
- 4,952
7
votes
2
answers
740
views
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 ...
- 151k
5
votes
1
answer
628
views
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.
...
- 461
33
votes
4
answers
3k
views
Does there exist 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 ...
- 151k
25
votes
1
answer
7k
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 frictionless.
...
- 151k
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
34
votes
6
answers
5k
views
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 ...
- 1,222
8
votes
1
answer
432
views
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 ...
- 9,720
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
6
votes
0
answers
450
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 ...
- 281
14
votes
2
answers
1k
views
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.
(source)
It was a good exercise, but at this point I ...
- 22.8k
22
votes
6
answers
15k
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 ...
- 15.4k
7
votes
1
answer
815
views
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 ...
- 151k
39
votes
3
answers
6k
views
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 ...
- 22.3k
38
votes
3
answers
4k
views
Parabolic envelope of fireworks
The envelope of parabolic trajectories from a common launch point is itself a parabola.
In the U.S. soon many will have a chance to observe this fact directly, as the 4th of July is traditionally ...
- 151k
3
votes
6
answers
8k
views
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 ...
- 33
12
votes
3
answers
3k
views
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 square ...
- 151k
13
votes
4
answers
1k
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 ...
- 151k
17
votes
6
answers
3k
views
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 ...
- 151k
24
votes
3
answers
3k
views
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 ...
- 13.2k
1
vote
0
answers
885
views
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?
- 121
16
votes
2
answers
4k
views
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 clay.
...
- 151k
15
votes
8
answers
2k
views
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 ...
- 5,243
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
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
9
votes
3
answers
354
views
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 ...
- 811