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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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 ...
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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 ...
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Reference for action-angle coordinates [closed]

Does anyone know a good reference to start studying Action-Angle coordinates? Thank you in advance !
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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 ...
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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 ...
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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 ...
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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 ...
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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}{\...
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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 ...
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The relation between unit quaternions and the virtual rotation of an angular velocity vector?

I am working with a calculus employed in multi rigid body dynamics problems introduced by professor E. Haug (c.f. this book). Let's define $\boldsymbol{\mathrm{e}} \in \mathcal{R}^4$ as a set of unit ...
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The derivation of thin plate spline interpolation energy function? [closed]

I am trying to derive the "thin plate energy functional". Given a thin plate $z = z(x,y)$, how does one derive easily the energy functional $$\iint_{\mathbb{R}^2} \,\left[\left(\frac{\...
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Applications of symplectic geometry to classical mechanics

It is claimed that classical mechanics motivates introduction of symplectic manifolds. This is due to the theorem that the Hamiltonian flow preserves the symplectic form on the phase space. I am ...
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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 ...
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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 ...
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Composite canonical transformation using Lie operators

Context: According to a Lie theorem in canonical transformation, if $f$ and $S$ are arbitrary functions of the canonical variable set $(\xi,\eta)$ (i.e. meaning that $\xi_{i},\eta_{i}$ are the 2n ...
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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 ...
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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 ...
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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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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\...
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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 ...
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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 ...
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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 $W(F)$...
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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" ...
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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 ...
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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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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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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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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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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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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 ...