Questions tagged [classical-mechanics]

Mathematics of classical mechanics, including Hamiltonian mechanics, Lagrangian mechanics, applications of symplectic geometry to mechanics, deterministic chaos, resonance etc.

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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 ...
Joseph O'Rourke's user avatar
12 votes
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388 views

A model of pillows

(The same system with slightly different questions has been asked in MSE.) Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \...
Daniel Castro's user avatar
11 votes
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222 views

Mathematical pendulum and $\mathbb C P^n$

I am very puzzled by the following remark on p.346 in Arnold's book "Mathematical methods of classical mechanics": Another method of construction the same symplectic structure on complex ...
Nikita Kalinin's user avatar
10 votes
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646 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 $\omega_{1}$...
Ali Taghavi's user avatar
9 votes
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365 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 ...
Joseph O'Rourke's user avatar
8 votes
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243 views

Billiards with incompatible regions

An existing question asks whether "almost every" two-dimensional billiard possesses at least one orbit that is dense in its interior. My question is about the following set of strong counter-examples:...
mjqxxxx's user avatar
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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 ...
Joseph O'Rourke's user avatar
7 votes
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101 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 $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...
Peter Heinig's user avatar
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7 votes
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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 ...
QuantumTheory's user avatar
7 votes
0 answers
160 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/...
Pedro Namtior's user avatar
6 votes
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156 views

Nonlinear-PDE arising from flat conformal Chebyshev nets

Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...
Daniel Castro's user avatar
6 votes
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234 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 ...
Yoav Kallus's user avatar
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6 votes
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Differential equation of line tangent to caustics

This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most ...
A B's user avatar
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5 votes
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Pocket billiards with balls in general position

There were at least two earlier MO questions about ideal pocket billiards. (Ideal: frictionless, perfectly elastic collisions.) Perfectly centered break of a perfectly aligned pool ball rack. Does ...
Joseph O'Rourke's user avatar
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239 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 (...
smanoos's user avatar
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On moments of inertia of planar and 3D convex bodies

The following observation can be readily proved using the perpendicular axes theorem and intermediate value theorem: "Given any planar figure C, through any point on it, there is at least one ...
Nandakumar R's user avatar
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Generalising Bäcklund transform to solve $\omega''(t)=t\sin\omega(t)$

Bäcklund transformations may be used also in ODE to solve non-linear problems; for instance, it's well known that for the equation $$ \frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=\sin\omega \tag{*}\label{...
Daniel Castro's user avatar
4 votes
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106 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. ...
Mehmet Ozan Kabak's user avatar
3 votes
0 answers
93 views

Mathematical formulation of beam: get stress/strain from forces and momentum

I'm working with static beams with Euler–Bernoulli model which ODE is $$ \dfrac{d^2}{dx^2} \left(EI \cdot \dfrac{d^2w}{dx^2}\right) = q(x). $$ With a beam along the $x$ axis, the solution consists of ...
Carlos Adir's user avatar
3 votes
0 answers
166 views

How to mix Lagrange mechanics + KKT conditions?

Question: How can I mix the concepts of Lagrange Mechanics and KKT conditions? I've learned that Lagrange Mechanics derivation comes from variational calculus, and in some formulations, we can add ...
Carlos Adir's user avatar
3 votes
0 answers
170 views

Non-linear, hyperbolic, 2nd order system of PDEs

This is a cross-post. In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system \begin{...
Daniel Castro's user avatar
3 votes
0 answers
102 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 ...
Oliver Nash's user avatar
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3 votes
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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 ...
Joseph O'Rourke's user avatar
3 votes
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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,...
Joseph O'Rourke's user avatar
3 votes
0 answers
166 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 ...
ZachMcDargh's user avatar
3 votes
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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 ...
Dayal C Strub's user avatar
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 ...
agt's user avatar
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3 votes
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557 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 ...
John Galt's user avatar
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Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials

I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses: Coulomb potential with a ...
michalt's user avatar
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Nonintegrable classical dynamical systems and deterministic chaos

I'm trying to delineate a minimal (and informal) "taxonomy" for classical continuous dynamical systems that could be interested by the phenomenon of "chaos" - unfortunately the ...
Lo Scrondo's user avatar
2 votes
0 answers
447 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 ...
R Mary's user avatar
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2 votes
0 answers
140 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 $xy$ ...
user135626's user avatar
2 votes
0 answers
51 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|...
Martin's user avatar
  • 153
2 votes
0 answers
163 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 ...
Austen's user avatar
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2 votes
0 answers
284 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 ...
agt's user avatar
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2 votes
0 answers
349 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 $...
hoj201's user avatar
  • 614
1 vote
0 answers
117 views

Integral expression for the Poisson bracket

I already asked this in the physics forum but without much attention, so I thought it might attract more attention here. Is there an integral expression for the Poisson bracket that can be derived ...
Nicolas Medina Sanchez's user avatar
1 vote
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123 views

Is it possible for the Lagrange multiplier to satisfy some constraints themselves?

I am using the field-theoretic langauge, so that we think of some action functional \begin{equation} S[f_l,T_{ij}]=\int_0^1 dt \int_{[0,1]^3}d^3\overrightarrow{x}\mathcal{L}(f_l(\overrightarrow{x}, t),...
Isaac's user avatar
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1 vote
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In the context of field-theoretic classical Lagrangian mechanics, can we choose the Lagrange multipliers to be time-independent? - from Physics SE

I originally posted this question on Physics SE, but I think it is more like a math question since I need rigorous justification. Could anyone please provide any insight to the below question: Let us ...
Isaac's user avatar
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1 vote
1 answer
234 views

Transport theorem in space craft control: tracking a reference angular velocity

I am reading the book named "Analytical mechanics aerospaces systems" by Schaub and Junkins. In section 7.2, the task is to control the spacecraft to track a specified angular velocity $w_r$ ...
sunxd's user avatar
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1 vote
0 answers
57 views

Vertical bundles of higher order tangent bundles

Let $M$ be a smooth (finite dimensional, Hausdorff and second countable) manifold. Let $T^kM$ be the manifold of equivalence class of curves that their derivates (in charts) agree up to order $k$. Let ...
alexpglez98's user avatar
1 vote
0 answers
121 views

Brachistochrone for a rolling sphere with slippage

I was recently looking into generalisations of the brachistochrone problem: for example, in this article the authors study the brachistochrone with Amontons-Coulomb friction where a bead slides along ...
Hollis Williams's user avatar
1 vote
0 answers
64 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 $W(F)$...
msaBU's user avatar
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1 vote
0 answers
60 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}...
user135626's user avatar
1 vote
0 answers
48 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,...
Tobsn's user avatar
  • 289
1 vote
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37 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{ \...
hoj201's user avatar
  • 614
0 votes
0 answers
61 views

Implications for a simple deterministic chaos definition

Among many others, one definition of deterministic chaos terms "chaotic" a classical dynamical system that satisfies the following three properties: sensitive dependence to initial ...
Lo Scrondo's user avatar
0 votes
0 answers
115 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, $X$ ...
Dash's user avatar
  • 1
0 votes
0 answers
177 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} },...
Narasimham's user avatar
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), $...
Jorge's user avatar
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