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4
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0answers
171 views

A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations ...
1
vote
1answer
52 views

One problem about tower stability [closed]

Some years ago i asked myself a question that I still can not answer. Here it is: A given tower consists of finite homogeneous cubic blocks staying one on another and equal to each other. What is ...
1
vote
0answers
57 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. ...
4
votes
3answers
2k 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 ...
26
votes
4answers
1k views

Stability of the Solar System

Is the Solar System stable? You can see this Wikipedia page. In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and ...
11
votes
0answers
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 ...
0
votes
1answer
55 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: ...
13
votes
4answers
2k views

What is the role of contact geometry in the hamiltonian mechanics?

Let us assume someone is interested in the study of Hamiltonian mechanics. What are good examples to illustrate him of the usefulness of contact geometry in this context? On one hand the Hamiltonian ...
4
votes
2answers
243 views

Derivative of eigenvectors of a matrix with respect to its components

Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as $$ B= \sum_{i=1}^3 ...
29
votes
5answers
4k views

Differentiable functions with discontinuous derivatives

For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...
8
votes
2answers
2k views

Classical Limit of Feynman Path Integral

I understand that in the limit that $\hbar$ goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation ...
5
votes
3answers
599 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 ...
5
votes
2answers
213 views

Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The ...
2
votes
1answer
179 views

Invariance of the Noether charge

The paper http://epubs.siam.org/doi/abs/10.1137/1023098 (Generalizations of Noether’s Theorem in Classical Mechanics, by Willy Sarlet and Frans Cantrijn) mentions "an interesting property of the ...
25
votes
1answer
868 views

Bouncing a ball down the stairs

In a nutshell, the question is whether it can be faster to bounce a ball down an infinite flight of stairs than to bounce it down a ramp with the same slope. To be more specific: this is a $2$ ...
86
votes
0answers
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 ...
7
votes
0answers
275 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 ...
3
votes
1answer
151 views

Local symplectomorphisms become global ones?

It is widely known that a local diffeomorphism is not necessarily a global diffeomosphism and so on. Now, I stumbled over the question whether in some particular cases, as I will describe below, ...
4
votes
1answer
236 views

Find a maximizing solution to an ODE which depends on a paramater function

(For the physical meaning of this problem see http://physics.stackexchange.com/questions/122818/how-should-i-throttle-my-rocket-to-reach-highest-altitude). Given $g \in (0,\infty), k \in C^1( [0, ...
14
votes
5answers
2k views

Can the equation of motion with friction be written as Euler-Lagrange equation, and does it have a quantum version?

My (non-expert) impression is that many physically important equations of motion can be obtained as Euler-Lagrange equations. For example in quantum fields theories and in quantum mechanics quantum ...
29
votes
3answers
3k views

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 directly, as the 4th of July is ...
1
vote
0answers
273 views

Proof of Arnold-Liouville theorem in classical mechanics [closed]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
7
votes
0answers
102 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 ...
0
votes
1answer
115 views

Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...
3
votes
1answer
168 views

higher order Noether identities

Noether's second variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities, which are relations among the Euler–Lagrange equations. How about relations ...
3
votes
1answer
973 views

1-jet bundle on vector bundle with metric connection

Background I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to ...
3
votes
2answers
229 views

Momentum a cotangent vector

Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities. Furthermore, many sources ...
3
votes
0answers
94 views

Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form ...
12
votes
2answers
3k views

*The* open problem in General Relativity?

Q. Is there a single, clear mathematical question that has emerged as the open problem in General Relativity? I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper, "Die ...
3
votes
0answers
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 ...
1
vote
0answers
106 views

Shortest rope to capture a sphere of diameter 1 [duplicate]

I have a perfect rigid sphere of diameter 1. I have infinite supply of rope. The rope is infinitely flexible and can be cut or glued without losing or adding length. The rope can be glued at any ...
11
votes
3answers
409 views

Orthogonal mud cracks and Maxwell's reciprocal figures

Is there a succinct mathematical/physical explanation of why mud cracks tend to meet orthogonally?                     Wikipedia image in this ...
3
votes
1answer
95 views

Stable equilibria of points on the 2-sphere

Suppose $n$ points lie on the sphere $S^2=\{x\in\mathbb{R}^3\mid \|x\|=1\}$ and are subjected to a repulsive acceleration that pushes away a point from each other point with an intensity proportional ...
2
votes
0answers
116 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 ...
3
votes
1answer
177 views

Weinstein's local classification of Lagrangian foliations

In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle. I ...
3
votes
0answers
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 ...
61
votes
5answers
17k views

The “Dzhanibekov effect” - an exercise in mechanics or fiction? Explain mathematically a video from a space station

The question briefly: Can one explain the "Dzhanibekov effect" (see youtube videos from space station or comments below) on the basis of the standard rigid body dynamics using Euler's equations? (Or ...
30
votes
1answer
3k views

Geometric interpretation of the half-derivative?

For $f(x)=x$, the half-derivative of $f$ is $$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x = 2 \sqrt{\frac{x}{\pi}} \;.$$ Is there some geometric interpretation of (Q1) this specific derivative, and, ...
6
votes
1answer
481 views

Is there a sideways-walking rolling convex body?

Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$. Place $K$ on an inclined plane, and let it roll down the plane, under some reasonable assumptions of friction between $K$ and the plane, ...
15
votes
6answers
2k 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 ...
21
votes
2answers
918 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 ...
3
votes
1answer
421 views

Planar linkage that traces a circle from its exterior?

Q. Is there a linkage in the plane that traces out a circle $C$ in such a manner that the interior of the disk bounded by $C$ is never intersected by any link througout the motion? What I ...
12
votes
9answers
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, ...
7
votes
2answers
205 views

Well-definedness of single-particle smooth billiards flow

Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well ...
9
votes
0answers
282 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 ...
3
votes
0answers
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 ...
8
votes
1answer
927 views

Oloid and sphericon: rolling develops entire surface

Wikipedia says that, "The oloid is one of the only known objects, along with some members of the sphericon family, that while rolling, develops its entire surface." Below are illustrations of ...
17
votes
4answers
1k views

Non-chaotic bouncing-ball curves

I was surprised to learn from two Mathematica Demos by Enrique Zeleny that an elastic ball bouncing in a V or in a sinusoidal channel exhibits choatic behavior:     (The Poincaré map ...
0
votes
2answers
772 views

Regarding understanding differential geometry [closed]

I am essentially looking for a book that would hold my hand through basic concepts to more complicated ones. I am coming from physics. I am looking to make some connections with Classical mechanics ...
34
votes
3answers
5k 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 ...