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
2
votes
0
answers
165
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 ...
- 1,038
4
votes
1
answer
396
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 ...
- 783
3
votes
0
answers
122
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 triangle,...
- 151k
4
votes
1
answer
372
views
Find a maximizing solution to an ODE which depends on a paramater function
(For the physical meaning of this problem see https://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, \...
- 141
25
votes
1
answer
3k
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 $2$ ...
- 12.5k
6
votes
1
answer
544
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, ...
- 151k
3
votes
1
answer
1k
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 mean ...
- 151k
48
votes
2
answers
7k
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, (...
- 151k
7
votes
2
answers
274
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 known;...
- 19.7k
33
votes
5
answers
12k
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 ...
- 19.7k
9
votes
0
answers
368
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 ...
- 151k
25
votes
5
answers
8k
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 ...
- 21.8k
9
votes
1
answer
3k
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 ...
- 151k
-1
votes
2
answers
1k
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 ...
user39066
22
votes
4
answers
2k
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 chaotic behavior:
(The Poincaré map ...
- 151k
3
votes
0
answers
167
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 ...
- 151
6
votes
1
answer
352
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 ...
- 10.8k
2
votes
1
answer
2k
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 http://arxiv.org/abs/...
- 23
3
votes
1
answer
275
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 ...
- 3,880
11
votes
5
answers
640
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 ...
- 151k
14
votes
1
answer
1k
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
...
- 151k
3
votes
0
answers
194
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
1
answer
2k
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 specify ...
6
votes
0
answers
237
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 ...
- 5,971
9
votes
2
answers
3k
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 ...
- 433
19
votes
3
answers
6k
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 ...
- 433
4
votes
1
answer
261
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 ...
user7807
1
vote
1
answer
1k
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 $\pi:(x_1,\xi_1,x_2,\xi_2)...
- 35
3
votes
2
answers
589
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.
...
6
votes
2
answers
3k
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 ...
- 463
9
votes
1
answer
559
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 ...
- 151k
6
votes
1
answer
1k
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 $V{<}...
- 4,306
0
votes
0
answers
187
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),
$...
- 59
2
votes
0
answers
285
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 ...
- 4,306
11
votes
3
answers
903
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 ...
- 151k
171
votes
8
answers
86k
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 ...
- 24.9k
5
votes
3
answers
3k
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 ...
- 4,306
2
votes
0
answers
356
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 $...
- 614
9
votes
1
answer
596
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 ...
- 657
3
votes
0
answers
174
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 ...
- 4,306
41
votes
2
answers
2k
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 $...
- 151k
5
votes
2
answers
233
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. ...
- 52.4k
15
votes
0
answers
517
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 ...
- 151k
20
votes
4
answers
3k
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 ...
- 4,306
37
votes
6
answers
3k
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 Law ...
- 151k
16
votes
5
answers
1k
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
1
answer
787
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 time,...
- 151k
3
votes
0
answers
559
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 ...
- 197
22
votes
2
answers
5k
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 ...
- 151k
19
votes
6
answers
3k
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?
- 921