All Questions
Tagged with classical-mechanics reference-request
26 questions
0
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0
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22
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Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?
I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
- 708
10
votes
1
answer
400
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Rigorous treatment of Ostrogradsky's instability theorem?
The Ostrogradsky instability theorem says that if a Lagrangian depends on more than the position and velocity, the corresponding Hamiltonian is unbounded below. This has been suggested as a reason why ...
- 1,924
1
vote
1
answer
330
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Global reduction of Hamiltonian with an integral of motion (Poincare' reduction)
This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question.
Background
I refer to the following concepts:
Liouville ...
- 469
7
votes
2
answers
2k
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Practical example of Hamiltonian reduction
I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p_1, \...
- 469
27
votes
4
answers
13k
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Hamiltonian, Lagrangian and Newton formalism of mechanics
If my thinking is wrong please let me know. I have little knowledge on beyond-college physics.
For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,...
- 8,071
3
votes
0
answers
135
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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
$$\ddot{...
- 41
3
votes
0
answers
179
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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 electrons ...
- 151k
2
votes
1
answer
527
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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 ...
- 16.5k
15
votes
4
answers
888
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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 ...
- 151k
9
votes
1
answer
3k
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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
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2
answers
1k
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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
14
votes
1
answer
1k
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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
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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 ...
6
votes
2
answers
3k
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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 ...
- 463
6
votes
1
answer
1k
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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
5
votes
3
answers
3k
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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 ...
- 4,306
41
votes
2
answers
2k
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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
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
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
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
11
votes
2
answers
1k
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 equal ...
- 151k
5
votes
1
answer
628
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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.
...
- 461
8
votes
1
answer
432
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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 ...
- 9,720
17
votes
5
answers
2k
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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 ...
- 151k
7
votes
1
answer
815
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 ...
- 151k