Skip to main content

Questions tagged [classical-mechanics]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
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 ...
4453's user avatar
  • 11
2 votes
1 answer
89 views

On the notation $ \vec F\cdot \mathrm ds $ for line integrals

Let $ \vec F\in \mathscr X(\mathbb R^3) $ be a vector field on the manifold $ \mathbb R^3 $ endowed with the Euclidean metric $ \mathrm ds^2 $. The inner product $ \vec F\mathop{\lrcorner} \mathrm ds^...
GeometriaDifferenziale's user avatar
2 votes
0 answers
38 views

Integrability of a constraint when $x$ and $y$ depend on $t$ [closed]

I'm working through problem 6 in chapter 1 in Goldstein's classical mechanics book. I've reduced it to asking, if $x$ and $y$ are coordinates and function of time $t$, whether the differential ...
CasualPhysicsEnjoyer's user avatar
1 vote
1 answer
58 views

$\ddot x$ vs. $\dot x^2$

I'm working on a physics assignment and am having some trouble. I need to integrate $r\dot\theta^2$ with respect to $t$. However, my trouble lies in the definition of the upper-dot format. Given: $$ \...
Chaserix's user avatar
1 vote
0 answers
58 views

particle in motion under the influence of friction

Let's consider a particle of mass = 1 Kg moving according to the law $$ \ddot x(t) = -V'(x(t))-\frac{2}{3}\dot x(t) = -x(t)^3+x(t)-\frac{2}{3} \dot x(t). $$ (The potential energy is $V(x)=\frac{x^4}{4}...
dattiluca's user avatar
2 votes
2 answers
225 views

Analytical solution of the equation for the launch angle of a projectile travelling the maximum trajectory length

I would like to find an analytical solution to the equation by which the launch angle is found for a projectile to travel the maximum trajectory length. Let $\theta$ be the angle at which the ball is ...
Bml's user avatar
  • 135
0 votes
0 answers
68 views

Question from an Old STEP paper

I found a question from a STEP paper from 1987, which is a preparatory high-level US equivalent exam (J87/P2/1). I have no idea to approach it. The diagram shows a light inextensible string with ...
skyfall's user avatar
  • 181
1 vote
1 answer
68 views

Calculate divergence $\nabla \cdot \mathbf{F}$ in spherical coordinates

I hope someone can clarify something for me. I'm looking at this problem: Calculate the integral $$ I = \int_{V} (\nabla \cdot \mathbf{F}) \ dv $$ where $\mathbf{F} = x \hat{\mathbf{x}} + y \hat{\...
math.lover's user avatar
0 votes
0 answers
45 views

Given an ellipse rolling without slipping over an inclined plane, determine the point in which occurs lose of contact with the support plane.

I want to be able to solve the problem of determining the movement of an ellipse that rolls without slipping on a curve with a given profile. For this reason it seems convenient to proceed in the ...
Cesareo's user avatar
  • 33.7k
2 votes
2 answers
63 views

Moment on manifold, no dependence on Lagrangian anymore?

I am a bit confused about the mathematics of classical mechanics. When working in $\mathbb{R}^n$, the generalized momentum is defined as the derivative of the Lagrangian with respect to the velocity ...
edamondo's user avatar
  • 1,327
0 votes
0 answers
50 views

What is the first integral of a Lagrangian system?

A question from Mathematical Methods of Classical Mechanics, Arnold, P90: Extend Noether's theorem to non-autonomous Lagrangian systems. There is a hint : Let $M_1=M\times\mathbb{R}$ be the ...
guoran guan's user avatar
0 votes
0 answers
35 views

$g^{\star}$ undefined terms in papers on geometric mechanics

I have been struggling for a long time to figure out the meanings of certain, related undefined terms in related papers on discrete geometric variational integrators. One example comes from Reference [...
Reb.Cabin's user avatar
  • 1,695
3 votes
1 answer
108 views

Writing a general solution to differential equation with Bessel functions

Consider a mass $m$ is placed on a horizontal level surface and attached to a spring, whose other end is attached to a vertical wall. The mass moves in a viscous medium, where the resistance force ...
gujaral's user avatar
  • 343
3 votes
1 answer
198 views

Why acceleration is not always parallel to velocity but velocity is always parallel to displacement?

Velocity is derivative of displacement : $$\vec v=\frac{\mathrm {d\vec r}}{\mathrm dt}$$ And acceleration is derivative of velocity. $$\vec a=\frac{\mathrm {d\vec v}}{\mathrm dt}$$ Given that their ...
An_Elephant's user avatar
  • 2,801
0 votes
0 answers
15 views

Law of the areas (2nd of Keplero) proof

I was reading the following proof of the law of the areas (the generalization for central forces): Consider a plane curve $t\mapsto (x(t),y(t))$, that in polar coordinates is given by $t\mapsto \rho(...
Luigi Traino's user avatar
0 votes
0 answers
87 views

How to make Peaucellier–Lipkin linkage work?

First: Don't crucify me for not "doing math", or asking a dumb question. School was a long time ago, and lose what you don't use. My current math knowledge ends with "rule of proportion&...
Bizz Keryear's user avatar
2 votes
1 answer
56 views

Equation of motion for Hamiltonian for n bodies.

Finding the arising equation of motion for given the Hamiltonian of n particles $$H = \frac{1}{2m} q_n^2+\frac{\alpha}{2}(y_{n+1}-y_n)^2+\frac{\beta}{4}(y_{n+1}-y_n)^4$$ The $\alpha,\beta, m$ are ...
unknown's user avatar
  • 391
1 vote
1 answer
40 views

Two Disagreeing Approaches to Applying Parallel Axis Theorem to a Rod

I've been staring at this simple rotational inertia problem for a while, but I keep getting two different answers. We have a uniform, thin rod with mass M and length L. Its pivot point is a distance L/...
Owen Matheson's user avatar
2 votes
0 answers
71 views

Constrained mechanical system as a limit of the motion under potential

In his Mathematical Methods of Classical Mechanics, V.I. Arnold states the following theorem without proof in pages 75-76: Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local ...
mcpca's user avatar
  • 352
1 vote
0 answers
20 views

Trying to understanding how different descriptions of the Legendre Transform are equivalent

Consider a convex function $f(x)$ and the line $y=px$. Let $x_p$ be the value of $x$ that maximizes $xp-f(x)$. This allows us to define the Legendre transform of $f$ as a function of $p$: $g(p) = x_pp ...
PhysicsIsHard's user avatar
1 vote
1 answer
84 views

Solving $\dot{x}^2+\frac{k}{m}x^2-\frac{k}{m}x^2_{max}=0$ from a spring oscillator problem

In my mechanics class we had a differential equation arise from a spring oscillator via its energy ($x^2_{max}$ is some constant maximum length) : $$\begin{align}\frac{1}{2}kx^2_{max}&=\frac{1}{2}...
AnthonyML's user avatar
  • 977
1 vote
1 answer
86 views

Time taken for a bead to slide down a wire.

A circular hoop of radius $r$ is stood vertically on a flat surface and held in place. A smooth wire is tightly stretched between the upmost point of the hoop and a point on the hoop at height $h$. A ...
Developer's user avatar
0 votes
1 answer
50 views

Why is the partial derivative of the generalized momentum with respect to generalized position equal to zero?

Let $q$ denote the generalized position vector, $v$ denote the generalized velocity vector and $p$ denote the generalized momentum vector. The Lagrangian of a mechanical system is a function of $q$, $...
PhysicsIsHard's user avatar
0 votes
0 answers
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 ...
Kian31's user avatar
  • 21
0 votes
0 answers
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'), ...
Davide Masi's user avatar
0 votes
0 answers
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. ...
Jonathan Zhang's user avatar
2 votes
1 answer
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 ...
Daigaku no Baku's user avatar
0 votes
0 answers
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 ...
Math_Physics's user avatar
1 vote
1 answer
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: $$F\left(t\right)=\begin{cases}A\mathrm{e}^{-\gamma t}\sin\left(\omega t\right),&t\ge0;\\0,&t<0.\end{cases}$$ ...
Andrés de Fonollosa's user avatar
1 vote
1 answer
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 ...
learningsheep's user avatar
1 vote
1 answer
77 views

Calculating the time taken for a bouncing ball to come to rest.

A ball is dropped from rest at a height $h_0$ and bounces from a surface such that the height of the $n$th bounce, $h_n$ is given by $h_n=αh_{n-1}$, where $h_{n-1}$ is the height of the previous, $(n-...
Developer's user avatar
4 votes
0 answers
163 views

Harmonic oscillator differential equation question

Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$ Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}....
Andrés de Fonollosa's user avatar
0 votes
0 answers
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 ...
Dasheng Wang's user avatar
2 votes
1 answer
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, ...
J.D's user avatar
  • 1,081
0 votes
1 answer
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 ...
Developer's user avatar
1 vote
1 answer
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 ...
Developer's user avatar
0 votes
0 answers
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 ...
ondrejkubu54gmailcom's user avatar
3 votes
1 answer
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 ...
tragus's user avatar
  • 31
0 votes
0 answers
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 ...
Andrea's user avatar
  • 77
1 vote
2 answers
34 views

An integral related to the theory of isotropic harmonic oscillator

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

Centroid of a parabolic arc

Find the centroid $C=(\bar{x},\bar{y})$ of the parabolic arc $y=16-x^2$ over $[-4,4]$. From symmetry, $$\bar{x}=0$$ To find $\bar{y}$, substitute $\tilde{y}=y$, $dL=\sqrt{1+4x^2} dx$ in $$\frac{\...
Starlight's user avatar
  • 1,774
0 votes
0 answers
33 views

Combining variables into differentials?

While studying a practice problem for a classical mechanics class, I encountered the following expressions in the solution given by my instructor: Having derived an equation for the upward vertical ...
OldWorldBlues's user avatar
0 votes
1 answer
78 views

Calculating the time of collision of two vehicles.

I have the following problem but I'm not sure how to get started. I have a car that is $14m$ behind a lorry. The car's initial velocity is $20 ms^{-1}$ and it is decelerating at $7.6 ms^{-2}$, whilst ...
Developer's user avatar
0 votes
2 answers
88 views

For non-square $n \times m$ matrices, when will $\| A \hat x\|$ be constant for all unit vectors $\hat x$?

For non-square $n \times m$ real matrices, when will $\| A \hat x\|$ be constant for all unit vectors $\hat x$? If $A$ is square, the answer is: If and only if $$A = \lambda I Q$$ where $\lambda$ is a ...
SRobertJames's user avatar
  • 4,412
1 vote
0 answers
57 views

Probability of collisions in the random n-body problem

Imagine we have n point particles in $\mathbb R^3$ of mass 1, with gravitational constant $G > 0$, moving according to Newtonian gravitation. Suppose we assign, to each initial position coordinate ...
user541020's user avatar
0 votes
0 answers
64 views

Rigid-body bars in a double pendulum setup

I've heard double pendulums are notoriously chaotic in their motion and that no closed-form solution to their ODEs of motion exist. Setting, by the theorem of kinetic energy $$\dot T= \Pi$$ I end up ...
Francesco Cereda's user avatar
0 votes
1 answer
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 ...
Olórin's user avatar
  • 5,455
-1 votes
1 answer
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" ...
mathlover123's user avatar
2 votes
0 answers
44 views

Question on notation in Mathematical Foundations of Elasticity (Marsden and Hughes)

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'...
Nate's user avatar
  • 801
0 votes
0 answers
87 views

Falling yo-yo: equation of motion, speed and change in momentum

The problem is: A yo-yo consists of two uniform discs, each of mass M and radius R, connected by a short light axle of radius a around which a portion of a thin string ...
Giulia Colzani 's user avatar

1
2 3 4 5
43