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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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 $w_r$ ...
1 vote
1 answer
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How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let $$ \begin{matrix} F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\ (x,z,p) \mapsto F(...
5 votes
1 answer
473 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" ...
19 votes
6 answers
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reference for Noether's theorem

What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?
39 votes
9 answers
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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\...
4 votes
1 answer
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Maximal length of trajectories in billiard

Consider discrete rectangular billard on lattice with integer dimensions a*b and n balls with radius $\frac{\sqrt 2}{2}$ and ...
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 ...
1 vote
1 answer
94 views

isotropy of the cotangent lift of a group action

I asked this question in stack exchange but have not received an answer, so I am posting it here. Given a group action on a manifold (e.g. configuration space of coordinates), cotangent-lift it to the ...
2 votes
0 answers
42 views

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 ...
12 votes
0 answers
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 \...
3 votes
2 answers
220 views

$2\mathrm{d}$ area maximizing short embeddings

Think of a beach ball on an pool of water or sand. Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a ...
2 votes
1 answer
151 views

Hyperbolic system of PDEs with elliptic-like boundary contions

Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 ...
10 votes
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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 $\omega_{1}$...
4 votes
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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 ...
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, (...
11 votes
3 answers
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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 ...
3 votes
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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 ...
3 votes
1 answer
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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 ...
2 votes
1 answer
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Mechanics: Model beam using differential vectorial formulation

At the Wikipedia there are the differential formulation for Euler-Bernoulli Beam \eqref{1} and Timoshenko Beam \eqref{2} $$ \begin{align} &\dfrac{d^2}{dx^2}\left(EI\dfrac{d^2w}{dx^2}\right) = q(x) ...
6 votes
0 answers
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. ...
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 ...
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),...
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 ...
13 votes
4 answers
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When sticks fall, will they weave?

Imagine $n$ $z$-vertical sticks uniformly spaced around a unit-radius circle in the $xy$-plane. At $t{=}0$, each is randomly $\epsilon$-perturbed from the vertical, and they fall under the influence ...
10 votes
1 answer
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Equations for an algebraic gömböc

A gömböc is a $3$-dimensional convex body (having uniform density) which has exactly one stable and one instable equilbrium position (see https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c). Such a convex ...
2 votes
0 answers
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Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach? [closed]

When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$ but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...
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 ...
4 votes
0 answers
231 views

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{...
2 votes
0 answers
101 views

How to quantify the non-commutativity of human body motion? [closed]

Some years ago, there was that question on this forum:"How to quantify noncommutativity?". I am asking that question in a context, human movement, which implies kinematic chains (like in ...
14 votes
6 answers
3k 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 ...
14 votes
2 answers
1k views

Polygonal billards programs

I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure. (source) It was a good exercise, but at this point I ...
9 votes
2 answers
371 views

Which convex bodies roll straight?

Let $K$ be a convex body in $\mathbb{R}^3$. Suppose $K$ is held at some position and orientation on an inclined plane, and released. Let there be sufficient friction so that it rolls without slippage. ...
9 votes
1 answer
633 views

Resources on screw theory in classical mechanics

I am considering a classical mechanics problem with a fairly complicated system where I think it might be possible to simplify the calculations using the formalism of screw theory and screw algebras, ...
169 votes
8 answers
83k 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 ...
25 votes
2 answers
2k views

Decidability of 3 body problem

Is there a result showing that something along the lines of the three body problem is undecidable? Or are they known to be decidable or neither? I mean problems along the lines of the following ...
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{...
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 ...
1 vote
1 answer
152 views

Derivative of eigenpair with respect to matrix

Suppose that $A$ is real and symmetric matrix (or tensor) of dimension $3 \times 3$, with its spectral decomposition $$A = \sum_{i=1}^3 \lambda_i\ n_i\otimes n_i$$ where $\lambda_i$, $n_i$ and $\...
0 votes
0 answers
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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 ...
1 vote
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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 ...
12 votes
3 answers
3k views

Generalizing square wheels rolling on inverted catenaries

It is not uncommon to see in a science museum a bicycle with square wheels that rides smoothly over a washboard-like surface made from inverted catenary curves (e.g., at the Münich museum). The square ...
0 votes
1 answer
156 views

Reference for action-angle coordinates [closed]

Does anyone know a good reference to start studying Action-Angle coordinates? Thank you in advance !
2 votes
0 answers
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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 ...
3 votes
1 answer
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Arnold's book on classical mechanics [duplicate]

Arnold's book “Mathematical methods of classical mechanics” develops the standard material on mechanics (e.g. the 3 Newton’s laws and the gravity law etc.). But what differs it from all other ...
4 votes
1 answer
297 views

Nonlinear ODE to linear PDE?

I am interested in when and how one can trade a non-liner ODE for a linear PDE. To explain what this could look like here is a physics-inspired discussion. Consider a classical mechanical system with ...
3 votes
4 answers
942 views

Applications of Hamiltonian formalism to classical mechanics

In many courses in theoretical classical mechanics Hamiltonian formalism takes an important place. However I did not see it applied to problems of classical mechanics (unless one expands the scope of ...
18 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 ...
17 votes
2 answers
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Example of ODE not equivalent to Euler-Lagrange equation

I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation $$\frac{\partial L}{\...
5 votes
0 answers
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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 ...
14 votes
1 answer
395 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. ...