8
votes
1answer
345 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 ...
-2
votes
2answers
540 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 ...
4
votes
0answers
458 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 ...
1
vote
0answers
120 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 ...
4
votes
2answers
597 views

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 ...
4
votes
1answer
518 views

How the Jacobi metrics may be useful in mechanics with or without constraints?

A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$ If ...
2
votes
3answers
968 views

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 ...
39
votes
2answers
1k views

Topple height of randomly stacked bricks

What is the expected height of a stack of unit-length bricks, each one stacked on the previous with a uniformly random shift within $\pm \delta$? The stack topples if the center of gravity of the top ...
28
votes
6answers
2k views

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 ...
8
votes
1answer
676 views

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 ...
10
votes
2answers
2k views

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 ...
10
votes
5answers
971 views

reference for Noether's theorem

What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?
10
votes
2answers
548 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 ...
5
votes
1answer
518 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. ...
7
votes
1answer
373 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 ...
12
votes
5answers
1k 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 ...
5
votes
1answer
499 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 ...