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# 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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### What is the definition of a Brachistochrone curve in a non-Euclidean space?

I have a problem where I have to study "the geometric properties of the Brachistochrone curve in non-Euclidean spaces". But I am confused about the definition of the Brachistochrone Problem/...
1 vote
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### 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
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### 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 ...
27 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 ...
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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 ...
135 views

### 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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### 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 ...
602 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 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{...
1 vote
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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 ...
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 ...
1 vote
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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 $W(F)$...
233 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" ...
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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 $2n$-...
1 vote
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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 ...