# 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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### 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(...

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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 ...

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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 ...

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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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### 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 \...

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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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### 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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### 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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### 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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### 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. ...

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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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### 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),...

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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 ...

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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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### 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 ...

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### 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 ...

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### 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.
...

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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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### 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 ...

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### 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, ...

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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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### 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 $\...

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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 $w_r$ ...

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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 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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### 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 ...