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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turbulence as an unsolved problem of classical mechanics
Why is it that turbulence is considered to be an unsolved problem of classical mechanics? What is meant by "unsolved"? Don't the Navier-Stokes equations apply to turbulent flows? It's difficult to ...
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1-jet bundle on vector bundle with metric connection
Background
I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to specify ...
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Elastostatics and homotopy type
In perfect elastostatics, the unknown is the displacement $x\mapsto y$, where $x\in\Omega\subset{\mathbb R}^3$ is the reference configuration, and $y\in{\mathbb R}^3$. It obeys to an 2nd-order PDEs. ...
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Poincaré Recurrence and Dense Sets
This is kind of a spin-off of the question asked here. Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The ...
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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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What are the canonical and earliest references to trivial symmetries in gauge systems?
I am trying to find canonical references and the history of trivial symmetries.
The earliest text book reference I can find is on page 69 of Quantization of Gauge Systems by Henneaux and Teitelboim.
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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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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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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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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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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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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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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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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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Minimizing action squared versus action
I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...
user7807
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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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Weinstein's local classification of Lagrangian foliations
In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I ...
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Find a maximizing solution to an ODE which depends on a paramater function
(For the physical meaning of this problem see https://physics.stackexchange.com/questions/122818/how-should-i-throttle-my-rocket-to-reach-highest-altitude).
Given $g \in (0,\infty), k \in C^1( [0, \...
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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 ...
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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{...
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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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Which motion is exclusive in 3D or higher dimensions?
Hi guys,
I have a simple question
Linear movement can be found in 1D, 2D and 3D world objects
Rotation can be found in 2D and 3D world objects.
Now, are there any kind of motion can only be found ...
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Functional Analysis and its relation to mechanics
Hi I'm currently learning Hamiltonian and Lagrangian Mechanics (which I think also encompasses the calculus of variations) and I've also grown interested in functional analysis. I'm wondering if there ...
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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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Herpolhode equation
Poinsot’s construction describes the motion of a freely rotating rigid body in terms of an ellipsoid rolling on a plane. (http://www.phys.ttu.edu/~huang24/Teaching/Phys5306/CH5C.pdf), and the path of ...
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$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 ...
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How to deal with the singular reduction of the Hamiltonian n body problem?
I would like to consider the reduced Hamiltonian $n$ body problem, but am struggling with the angular momentum reduction seeing as the $SO(3)$ action is not free and the reduction is singular.
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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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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 ...
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higher order Noether identities
Noether's second variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities, which are relations among the Euler–Lagrange equations.
How about relations ...
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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).
$$
With a beam along the $x$ axis, the solution consists of ...
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How to mix Lagrange mechanics + KKT conditions?
Question: How can I mix the concepts of Lagrange Mechanics and KKT conditions? I've learned that Lagrange Mechanics derivation comes from variational calculus, and in some formulations, we can add ...
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Non-linear, hyperbolic, 2nd order system of PDEs
This is a cross-post.
In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system
\begin{...
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Arnold's book on classical mechanics [duplicate]
Arnold's book “Mathematical methods of classical mechanics” develops the standard material on mechanics (e.g. the 3 Newton’s laws and the gravity law etc.). But what differs it from all other ...
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Sampling point from the surface of an n-dimensional ellipsoid with uniform distribution
I am wondering if exist an efficient computational method for sampling points belonging to the surface of an ellipsoid in $n$-dimensional space with n even, I am thinking in the phase space of a ...
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Is there any connection between Lagrange points and the icosahedron?
Given the Newtonian two-body problem, one can ask if there are any orbits that allow a test particle to maintain a fixed configuration relative to the two bodies. In other words, in a frame that ...
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Motivation for the existence of periodic solutions [closed]
I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
$$\ddot{...
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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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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,...
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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 ...
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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 ...
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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 ...
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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 ...
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Inverse problem of the calculus of variations for autonomous second-order ODEs
Consider the following particular case of the inverse problem of the calculus of variations: given a system of second-order equations
$$
\ddot{q}^i = f^i(q, \dot{q}, t), \quad i = 1, \dots, n, \label{...
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Hyperbolic system of PDEs with elliptic-like boundary contions
Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 ...
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Mechanics: Model beam using differential vectorial formulation
At the Wikipedia there are the differential formulation for Euler-Bernoulli Beam \eqref{1} and Timoshenko Beam \eqref{2}
$$
\begin{align}
&\dfrac{d^2}{dx^2}\left(EI\dfrac{d^2w}{dx^2}\right) = q(x) ...
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Derivation of Bessel functions
I am writing a summary on a work on Fluid Dynamics that develops irrotational flow states that appear to interact amongst each other according to the equations of Electromagnetism http://arxiv.org/abs/...
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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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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 ...
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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 ...
