# Questions tagged [classical-mechanics]

For questions on classical mechanics from a mathematical standpoint. This tag should not be the sole tag on a question.

2,103 questions
Filter by
Sorted by
Tagged with
1 vote
30 views

### Action-angle variable construction and the assumption of independent actions

Let $N$ be an $n$-dimensional smooth manifold, $M=T^*N$, and $\omega$ its canonical symplectic structure. Any $H\in\mathcal{C}^\infty(M)$ induces a Hamiltonian vector field $X_H$ defined implicitly by ...
• 11
89 views

34 views

### For the following setup, what is the second holonomic constraint to limit the bead to one dimension?

We have a bead sliding along a smooth wire in the shape of a cycloid with equations, $$x=a(\theta-\sin\theta),\space y=a(1+\cos\theta),$$where $0\leq\theta\leq2\pi$. To find the number of ...
• 21
35 views

### An equivalent condition for isometries

Despite the long physical introduction, I swear this is a mathematical question. While introducing the motion of a point seen by two observers O and O' (with respective Euclidean spaces E3 and E3'), ...
34 views

### Formal adjoint of the symmetric gradient operator; is there a good interpretation?

Some context: I am reading Strang's paper, A Framework for Equilibrium Equations. In it, he gives a simple example with beams and rods. I'm trying to reproduce his arguments for 2D linear elasticity. ...
59 views

### Jacobi identity for Poisson bracket in local coordinates

Suppose a bivector field $\pi^{ij}$ such that $\pi^{ij}=-\pi^{ji}$, $\pi^{ij}\partial_{i}f\partial_{k}g=\{f, g\}$ defines a Poisson bracket $\{,\}$ on a smooth manifold (Einstein's summation is ...
• 1,177
63 views

### One mechanical question, two possible solutions. Can mathematics help to predict the actual physical event?

I have tried on my kid's textbook question (so to have meaningful discussion with him), but it seems to have two solutions: My first calculation is as below: . Second solution: . Both calculated ...
1 vote
85 views

### Solving the differential equation $m\frac{d^2x}{dt^2}=Ae^{-\gamma t}\sin(\omega t).$

A particle of mass $m$ moving along the $x$-axis, is acted upon by a force: $$F\left(t\right)=\begin{cases}A\mathrm{e}^{-\gamma t}\sin\left(\omega t\right),&t\ge0;\\0,&t<0.\end{cases}$$ ...
1 vote
42 views

### Using integration to solve non-constant acceleration problems

The question is: A body moves along a straight line with acceleration $a$ in meters per second squared given by $a = \frac{7}{36}t$ where $t$ is the time in seconds. Initially the body is at rest at ...
1 vote
77 views

37 views

### Introducing internal spin into the mechanical equations of an orbiting particle

In the preface of Lawson & Michelsohn's Spin Geometry, they mentioned some physics relevant to their book which I find interesting: The theory of Dirac had another interesting feature. In the ...
• 929
53 views

### Arclength parametrisation in 4D.

My question is to do with parametrisation of arclengths. As part of a course on mechanics (with mathematical focus), I have covered intrinsic coordinates in a plane curve. In class and homeworks etc, ...
• 1,081
95 views

### Adding mass to a hemispherical bowl until it sinks

I've been struggling with this question, which would probably be more appropriate to ask on Physics Stack Exchange which I've already done, but am hoping for some guidance here too. A hemispherical ...
1 vote
46 views

### Finding angle relative velocity makes with unit vector i.

I've attempted this question but can't seem to get the right answer. Three identical particles A, B and C are moving in a plane and, at time $t$, their position vectors, $â$, $b̂$ and $ĉ$, with ...
69 views

### Lift of a Riemannian metric to the cotangent bundle

Let us have a Riemmanian manifold $(M,g)$. Is there some “natural“/canonical way to lift the metric to the cotangent bundle $T^* M$, i.e. define a metric on $T^*M$? I ask because I have read some ...
120 views

### Circle rolling between two functions

Consider a circle of radius $r_s$ that is tangent to two curves $r(\theta)$ and $R(\theta)$ at points $E_1, E_2$ respectively, defined in polar coordinates. Knowing the function $r(\theta)$, find the ...
• 31
25 views

### Order of derivation if calculus of variation

In reading the derivation of the relativistic free particle (covariant) equation of motion by extremising the relativistic action $-mc\int_A^B \, ds$, I stumbled upon a passage where the author ...
• 77
1 vote
34 views

### An integral related to the theory of isotropic harmonic oscillator

The classical Hamiltonian for an isotropic harmonic oscillator in polar coordinates is \tag{1} H = \frac{1}{2m}\left(p_\rho^2 + \frac{p_\varphi^2}{\rho^2}\right) + \frac{1}{2}m\omega^2\...
• 1,084
1 vote
78 views

49 views

### How to use Euler-Lagrange equation to model a linear particle?

Suppose I have a particle with mass $m$ situated on a frictionless line which we can model using $\mathbb{R}$. Suppose that we supply/push the particle $m$ with a force $f(t)$. Let the position of ...
• 5,455
81 views

### Preserving quantities under linear transformations. [closed]

Find the linear transformation between two coordinate systems on a two-dimensional flat surface that preserves the quantity $s^2 = x^2 − y^2$. I can't figure out what it means to "preserve" ...
I have a small question regarding the usage of notation in this book. It is defining differential forms and their properties in its section, but the $\mid$ operator is being used in a way that I haven'...