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3
votes
1answer
117 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 ...
25
votes
2answers
1k 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
2answers
147 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 ...
25
votes
5answers
1k 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 ...
7
votes
0answers
159 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 ...
8
votes
4answers
562 views

Can the equation of motion with friction be written as Euler-Lagrange equation?

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 ...
0
votes
0answers
120 views

what aspects of Lagrangian submanifolds make them important? [duplicate]

what aspects of lagrangaian submanifolds make them important to concern, and for what reason we call them "lagrangian", are they related to lagrangian function? I maen, I wanna know the physical ...
8
votes
1answer
247 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 ...
0
votes
0answers
48 views

Significance of a combination of Weierstrass zeta values arising in the Lagrange top

I'll follow the notation of Whittaker and Watson. Let $\zeta(z)$ be the Weierstrass zeta function with half periods $\omega_{1,2}$ and $\eta_i=\zeta(\omega_i)$. I am interested in the case where the ...
-2
votes
2answers
500 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 ...
17
votes
4answers
645 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 PoincarĂ© map ...
2
votes
0answers
97 views

How to find solutions of non-linear ODE with particular BCs

What are some methods, numerical or otherwise, of finding solutions to nonlinear ODEs that satisfy particular boundary conditions? In particular, I'm looking for curves y(s) constrained to a ...
4
votes
1answer
188 views

Relation between Lee and Yang' s “circle theorem”, zeta functions and Weil conjectures?

Ruelle mentions ( http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B94%5D.pdf ) Lee and Yang' s "circle theorem", which comes from statistical mechanics and shall have not yet explored connections with zeta ...
2
votes
1answer
468 views

Derivation of Bessel functions

I am writing a summary on a work on Fluid Dynamics that develops irrotational flow states that appear to interact amongst each other according to the equations of Electromagnetism ...
0
votes
0answers
155 views

Variational Problem v.s. Initial Value Problem

Is there a way to relate the variational problem where one specifies $x$ initially and finally to the initial value (Cauchy) problem where one specifies both $x$ and $p$ initially? As an example, ...
11
votes
5answers
492 views

To what extent does trajectory determine gravity sources?

Suppose one has in-hand an accurate time-space trajectory in $\mathbb{R}^3$ of a (small) body, say an asteroid or satellite—effectively a point. To what extent does this trajectory determine the ...
4
votes
0answers
429 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
117 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 ...
5
votes
0answers
171 views

Generalization of the non-existence of a monostatic planar body

Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only one orientation of stable equilibrium and one orientation of unstable ...
5
votes
2answers
1k views

Classical Limit of Feynman Path Integral

I understand that in the limit that h_bar 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 ...
10
votes
2answers
2k views

Classical Limit of Quantum Mechanics

There is a well-known principle that one can recover classical mechanics from quantum mechanics in the limit as $\hbar$ goes to zero. I am looking for the strongest statement one can make concerning ...
4
votes
1answer
219 views

Minimizing action squared versus action

I have a very basic question in the calculus of variations: Suppose I want to minimize the functional $$A[r, r'] = \int_\Omega L(r, r') dx $$ When is it possible to say that extremals of $A$ agree ...
1
vote
1answer
310 views

Generating functions and Lagrangian submanifolds

I'm interested in showing the existence of a generating function. Explicitly: Suppose $\Lambda\subset T^*M\times T^*M$ is a Lagrangian submanifold. Consider the projection ...
3
votes
2answers
385 views

How to deal with the singular reduction of the Hamiltonian n body problem?

I would like to consider the reduced Hamiltonian $n$ body problem, but am struggling with the angular momentum reduction seeing as the $SO(3)$ action is not free and the reduction is singular. ...
4
votes
2answers
547 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 ...
7
votes
1answer
284 views

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 ...
4
votes
1answer
471 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 ...
0
votes
0answers
165 views

Transformation of the dynamics of mechanical system under coordinate change

It is well known that the dynamics equation for a mechanical system (e.g. a robotics manipulator) is given be the Euler-Lagrange equation which takes the particular form (in the simplified version), ...
2
votes
0answers
210 views

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 ...
10
votes
3answers
617 views

“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 ...
46
votes
4answers
8k 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 ...
2
votes
3answers
908 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 ...
1
vote
0answers
237 views

Dissipative Hamiltonian System with a Periodic Force

Let $H:P \to \mathbb{R}$ be a Hamiltonian on a symplectic manifold $(\omega,P)$ and let $X_H: P \to TP$ be the Hamiltonian vector-field. Let $F:P \to T^*P$ be a dissipative force field such that for ...
9
votes
1answer
402 views

Classical analogue of the Stone-von Neumann Theorem?

Let $U_s$, $V_t$ be a pair of continuous $n$-parameter groups ($n < \infty$) of unitary operators on a complex Hilbert space $\mathcal{H}$. The Stone-von Neumann Theorem establishes that any such ...
2
votes
0answers
131 views

What happens when Appell-Chetaev's rule for constrained mechanical systems is not applicable?

Background: Let be given a mechanical system whose configuration space is a manifold $Q$, and the kinetic energy is a metric $K$ on $Q$, in presence of a potential function $V$. Let us identify the ...
38
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 ...
4
votes
2answers
213 views

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. ...
10
votes
0answers
361 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 ...
12
votes
4answers
1k 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 ...
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 ...
15
votes
5answers
912 views

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 ...
8
votes
1answer
671 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 ...
3
votes
0answers
486 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 ...
10
votes
2answers
1k 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
938 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?
31
votes
2answers
1k 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 ...
7
votes
1answer
658 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 $\hat g := \sqrt g$ ...
20
votes
2answers
783 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 ...
18
votes
1answer
675 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:           ...
2
votes
2answers
370 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 ...