# Tagged Questions

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### References for the Poincaré-Cartan forms

Hello, everybody. I'm looking for some reference about the Poincaré-Cartan form, I do not know how it is defined, I just know that it is used in Lagrangian mechanics but I have not found any ...
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### Generalizing a square wheel to a body rolling on a surface

A square wheel rolling on a catenary road maintains the wheel center at a fixed height, a well-known construction previously discussed on MO (e.g., "Generalizing square wheels rolling on inverted ...
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### 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 ...
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### “Rolling Geodesics”: Designing a $k$-putt green

I am interested in what might be called rolling geodesics, paths of physical particles confined to a surface in $\mathbb{R}^3$ under certain force conditions. Here I will pose a specific (but ...
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### 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 ...
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### The Lagrangian formulation of mechanics without going through variational principles.

In some texts on classical mechanics and not only, the Euler--Lagrange equations of motion are directly obtained as solution of variational problems. On the other side, sometimes reading about ...
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### Elastostatics and homotopy type

In perfect elastostatics, the unknown is the displacement $x\mapsto y$, where $x\in\Omega\subset{\mathbb R}^3$ is the reference configuration, and $y\in{\mathbb R}^3$. It obeys to an 2nd-order PDEs. ...
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### 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 ...
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### 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 ...
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### Billiard dynamics under gravity

Has the dynamics of billiards in a polygon subject to gravity been studied? What I have in mind is something like this:            Still Snell's ...
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### G-bundles in classical mechanics

The paper Geometry of the Prytz Planimeter described a mechanical instrument whose configuration space is an $S^1$-bundle with an $SU(1,1)$ action. That paper goes on to study the holonomies of ...
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### The rain hull and the rain ridge

Rain falls steadily on an island, a 2-manifold $M$, which you may assume, as you prefer, is: (a) smooth, or (b) a PL-manifold, or perhaps even (c) a triangulated irregular network (TIN). After a time,...
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### 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 ...
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### Surface equivalent of catenary curve

A catenary curve is the shape taken by an idealized hanging chain or rope under the influence of gravity. It has the equation $y= a \cosh (x/a)$. My question is: What is the shape taken by an ...
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### reference for Noether's theorem

What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?
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### 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 $R \subseteq \mathbb{R}$...
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### 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 $\hat g := \sqrt g$ ...
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### 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 ...
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### 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:           ...
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### 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 ...
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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 the ...
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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 counter-examples:...
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### 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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### 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 equal ...
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### 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 $1/n$...
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### 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 ...
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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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### 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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### $\exists$ 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...