Questions tagged [classical-mechanics]

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7
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0answers
108 views

Hanging a cube with string

This is a variation on a (much) earlier MO question, Hanging a ball with string. Here instead the task is to arrange a net of string to hang a unit cube. Assume: The string is inelastic. There is no ...
4
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1answer
136 views

history of geometric mechanics

I was thinking about the foundations of geometric mechanics and its precursors. I wondered who was the first to realized the equivalence between Riemannian geometry and Lagrangian mechanics. In ...
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0answers
51 views

Proove the equivalence of two Lagrangians

Consider Lagrangian $$ L_1= q\dot{\alpha}+\alpha^2 $$ The Euler-Lagrange equations for $q$ and $\alpha$ read $$ \dot{q}=2\alpha $$ $$ \dot{\alpha}=0 $$ These two equations can be combined, ...
36
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7answers
3k views

Interpretation of the action in classical mechanics

In classical mechanics the dynamics on a manifold $M$ are characterised by the minimisation of a functional $$ \min_{q \in C^\infty(\mathbb{R},M)} \int_{\mathbb{R}}L(q(t),\dot{q}(t))dt, $$ where $L:TM\...
6
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2answers
213 views

Movement of repelled particles in a ball

EDIT: Given a system of $N\geq 3$ charged point particles in $\mathbb{R}^3$ of the same charge which interact according to Coulomb law (thus they repell one from each other). Is it possible that ...
8
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4answers
2k views

mathematical physics without partial derivatives

Remark: All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions ...
1
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0answers
38 views

Elasticity tensor in terms of principal stretches

Suppose we are given a frame-indifferent isotropic function $W:GL_+(3) \to [0,\infty)$, where $GL_+(3)$ denotes the set of all real $(3\times 3)$-matrices with positive determinant. We can write $W(F)$...
2
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0answers
77 views

Definition of a moment map with physical context

This was originally posted on Math Stack Exchange, but without an answer. I thus move it here, and hope it's not because I express it unclearly. Suppose $(M,\omega)$ is a symplectic manifold "well" ...
30
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3answers
2k views

Why is the billiard problem for obtuse triangles so hard?

This is an incredibly naive question so this may be closed. Nevertheless, I have been reading about the problem asking if every obtuse triangle admits a periodic billiard path, which has been open ...
2
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0answers
130 views

How to check conditions for Liouville-Arnold theorem? [closed]

Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem: Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...
1
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1answer
221 views

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 ...
6
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2answers
462 views

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, \...
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0answers
75 views

Deformation gradient conservation law from Lagrangian to Eulerian formulation

In the following, I use the standard notation for (solid) mechanics and conservation laws, i.e. $F$ the formation gradient, $H$ the cofactor, $v$ the velocity field and $J$ the Jacobian. Moreover, $X$ ...
29
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5answers
8k views

Six yolks in a bowl: Why not optimal circle packing? [closed]

Making soufflé tonight, I wondered if the six yolks took on the optimal circle packing configuration. They do not. It is only with seven congruent circles that the optimal packing places one in the ...
10
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3answers
579 views

Navier-Stokes fluid dynamics, Einstein gravity and holography

There was some activity a while ago, like 10 years ago, string theoreists try to relate the fluid dynamics, for example, governed by Navier-Stokes equation, to the Einstein gravity, and its ...
0
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0answers
163 views

Towards recognizing St. Venant geometrical invariant

Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates: $$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} },...
8
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0answers
373 views

Determinant as a Hamiltonian

Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...
22
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1answer
473 views

Tying knots via gravity-assisted spaceship trajectories

Q. Can every knot be realized as the trajectory of a spaceship weaving among a finite number of fixed planets, subject to gravity alone?           To make this more ...
4
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1answer
197 views

symplectic topology of (perturbed) KAM tori

Consider a real analytic $H_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies ...
6
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0answers
84 views

Is there a convex three-dimensional body with constant width and only finitely-many equilibria? Or: do spheroform gömböcök exist?

Mathematical questions. The mathematical (and 'gravity'-free) formulation of the question in the title is given by the following questions: Q1. Does there exist $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...
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0answers
208 views

How to make sense of the Euler Lagrange equations for an infinite action?

The Euler–Lagrange equation is an equation satisfied by a function $q$, which is a stationary point of the functional $S(\boldsymbol q) = \int_a^b L(t,q(t),\dot{q}(t))\, \mathrm{d}t$ Say we have an ...
3
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2answers
296 views

Can one obtain this ODE as an Euler-Lagrange equation?

Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...
10
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1answer
279 views

Conjecture: Finitely many points where gravitational field due to N masses vanishes

Given a configuration $C$ of $N$ distinct fixed points of equal mass in the plane (eventually in space), let $f_C(N)$ denote the number of points $P$ for which the gravitational field at $P$ vanishes. ...
4
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2answers
168 views

A Stochastic Dynamical Billiard

Consider the following stochastic dynamical system. Fix $a > 0$, $b > 0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t))$ be the position at time $t$ of a point which moves in the rectangle ...
1
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0answers
48 views

Optimal contour shape for variational problem over captured area

Let's assume we have a continuous and finite scalar function $f(x,y)$ over the $xy$ plane ($\mathbb{R}^{2}$) and this function is to be integrated over a bounded area (surface) $A\subset\mathbb{R}^{2}...
2
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0answers
121 views

Formulation of contour variational problem

I am having difficulty formulating a problem, which involves optimizing a contour shape, into a well-posed variational form that would give a reasonable answer. Within a bounded region on the $xy$ ...
13
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3answers
431 views

Random N-body problem

Suppose there are $N$ unit-mass particles whose initial positions are uniformly distributed in a unit-radius disk. Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...
32
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3answers
3k views

Do bubbles between plates approximate Voronoi diagrams?

For example, soap bubbles:                   Image from UPenn: "A 2-dimensional foam of wet soap bubbles squashed between glass plates, after 10 hours ...
2
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1answer
364 views

Sampling point from the surface of an n-dimensional ellipsoid with uniform distribution

I am wondering if exist an efficient computational method for sampling points belonging to the surface of an ellipsoid in $n$-dimensional space with n even, I am thinking in the phase space of a ...
22
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4answers
11k views

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,...
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0answers
41 views

Negative Definiteness of Hopf-Lax-Oleinik Semigroup

Denote by $H_{t}$ the Hopf-Lax semigroup, i.e.\begin{equation} H_{t}f(x)=\inf_{y\in\mathbb{R}}\left\lbrace f(y)+\frac{(x-y)^{2}}{2t}\right\rbrace.\end{equation} Is $H_{t}$ negative definite on bounded,...
1
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1answer
70 views

an example on kinematical theory in the book “A treatise on the analytical dynamics of particles and rigid bodies”

In page 3, Example 1 says "A lamina moves in any manner in its plane. Prove that the locus at any instant of points which are at inflexions of their paths is a circle, which touches the loci in the ...
8
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1answer
452 views

When does a Lagrangian dynamical system have an equivalent Hamiltonian description?

Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$ (i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q_{i},\dot{q}_{i},t)$, $i=1,2,......
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0answers
49 views

From Boundary to righthandside

I have a problem coming from linear elasticity in $(x,y,z)\in\mathbb{R}^2\times \mathbb{R}^+$, $t\in \mathbb R$: $$\left\{\begin{aligned}\partial_{tt} \sigma&=A(D_x,D_y,D_z) \sigma\\ \sigma\big|...
9
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1answer
346 views

Ping-pong progress through a quincunx

A quincunx or Galton board consists of staggered pegs from which ping-pong balls bounce and eventually display a binomial / normal distribution in catch-bins. I am wondering if the downward progress ...
21
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1answer
1k views

Why are Lagrangian submanifolds called Lagrangian?

Much of the terminology in symplectic geometry comes from classical mechanics: the symplectic manifold is modeled on a cotangent bundle $T^*N$ of some configuration space $N$ with local position ...
17
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2answers
462 views

Construction of an optimal electron cage

I will describe the question first in 2D, but my interest is in $\mathbb{R}^3$. An electron $x$ will shoot from the origin along an initial vector $v$. You know the speed $|v|$ but not the direction. ...
3
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0answers
93 views

Is there any connection between Lagrange points and the icosahedron?

Given the Newtonian two-body problem, one can ask if there are any orbits that allow a test particle to maintain a fixed configuration relative to the two bodies. In other words, in a frame that ...
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0answers
33 views

The isotropy group for the Euler-Lagrange vector-fields

Let $Q$ be a manifold, and let $X_{EL}$ be a second order vector-field on $TQ$ derived from the Euler-Lagrange equation, $$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q} } \right) - \frac{ \...
4
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0answers
232 views

A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (...
1
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1answer
90 views

One problem about tower stability [closed]

Some years ago i asked myself a question that I still can not answer. Here it is: A given tower consists of finite homogeneous cubic blocks staying one on another and equal to each other. What is ...
4
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0answers
83 views

Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well. ...
0
votes
1answer
67 views

underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: ...
9
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2answers
3k views

Derivative of eigenvectors of a matrix with respect to its components

Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as $$ B= \sum_{i=1}^3 \lambda_{i}(...
8
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2answers
285 views

Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The ...
7
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0answers
408 views

Question about theorem in Arnold's book on action-angles variables

I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question) If you don't have the book or ...
3
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1answer
231 views

Local symplectomorphisms become global ones?

It is widely known that a local diffeomorphism is not necessarily a global diffeomosphism and so on. Now, I stumbled over the question whether in some particular cases, as I will describe below, ...
27
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4answers
2k views

Stability of the Solar System

Is the Solar System stable? You can see this Wikipedia page. In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and ...
2
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0answers
916 views

Proof of Arnold-Liouville theorem in classical mechanics [closed]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
7
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0answers
144 views

Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes. (An example:https://www.youtube.com/...