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
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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 $2n$-...
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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 ...
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2
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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, \...
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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, $X$ ...
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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 ...
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3
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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 ...
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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} },...
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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}$...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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. ...
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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 ...
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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}...
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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$ ...
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3
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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 ...
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33
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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 ...
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3
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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 ...
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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,...
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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,...
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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 ...
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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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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|...
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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 ...
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24
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answer
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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 ...
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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.
...
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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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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,
$$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q} } \right) - \frac{ \...
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A soft question on Gauge Equivalence in Integrable Systems
I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (...
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One problem about tower stability [closed]
Some years ago i asked myself a question that I still can not answer. Here it is:
A given tower consists of finite homogeneous cubic blocks staying one on another and equal to each other. What is ...
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Dynamics of pairwise distances in the $n$-body problem
Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
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underdamped oscillation with quadratic decay
I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form:
...
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Derivative of eigenvectors of a matrix with respect to its components
Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as
$$
B= \sum_{i=1}^3 \lambda_{i}(...
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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 ...
user60504
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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 ...
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3
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1
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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, ...
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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 ...
user68208
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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 ...
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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/...
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Lagrangian flow preserves symplectic form
Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...
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Momentum a cotangent vector
Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.
Furthermore, many sources ...
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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{...
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*The* open problem in General Relativity?
Q. Is there a single, clear mathematical question that has emerged as
the open problem in General Relativity?
I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper,
"Die ...
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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 ...
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2
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1
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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 ...
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Shortest rope to capture a sphere of diameter 1 [duplicate]
I have a perfect rigid sphere of diameter 1.
I have infinite supply of rope. The rope is infinitely flexible and can be cut or glued without losing or adding length. The rope can be glued at any ...
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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 ...
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4
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1
answer
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Stable equilibria of points on the 2-sphere
Suppose $n$ points lie on the sphere $S^2=\{x\in\mathbb{R}^3\mid \|x\|=1\}$ and are subjected to a repulsive acceleration that pushes away a point from each other point with an intensity proportional ...
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