1
vote
2answers
34 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
3
votes
4answers
745 views

A 6 meter ladder…

A $6$ meter long ladder leans with a vertical wall and top of the ladder is 3 meters above the ground.If it slips at a rate of $2$ m/s then how fast the level is decreasing from the wall? My ...
1
vote
2answers
22 views

The volume is preserved by the flow: where is the absolute value?

Consider the following excerpt of the Liouville's theorem proof taken from "Arnold - mathematical methods of classical mechanics": In changing the variables in the integral, I don't understand why ...
0
votes
0answers
33 views

Is this divergence-free? (Double Pendulum)

Concerning this page http://scienceworld.wolfram.com/physics/DoublePendulum.html for the double pendulum the moving equations are given by $$ ...
1
vote
1answer
57 views

Calculating a double pendulum

consider the following situation of a double pendulum. We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\ ...
1
vote
1answer
30 views

What is a homographic solution in three body problem?

I came across Saari's homographic conjecture in Three Body problem. I need more information on what exactly is a homographic solution and how is it different from a homothetic solution?
3
votes
0answers
47 views

Door mechanism differential equation

I have been wondering about a door mechanism I have seen. It has a wire attached to the upper corner of the door and from there to the corresponding corner in the door frame, where a weight hangs from ...
1
vote
4answers
148 views

Mathematical modelling that involves projectile motion

I was asked to solve a mathematical differential equation to find the time taken by an object to reach the highest point and the time taken by the object to fall from its highest point to ground. I ...
2
votes
2answers
46 views

What is the answer to $\int x(t)dt$?

$\int x(t)dt$? I'm trying to solve a differential equation, but I've hit a strange brick wall that I never used to have a problem climbing over. This question is about mechanics & the equation ...
0
votes
1answer
24 views

Vector differential equation problem

I was trying to do this question from a past paper, but I'm not sure how to proceed. The question is: A particle of mass $m$ moves subject to a force $\mathbf F = A(y\mathbf i + x\mathbf j)$ where ...
1
vote
0answers
51 views

Linearization of Implicit ODE (Equations of Motion)

let's say we have a system with vector $q_{(t)}$ representing the degrees of freedom (DoF), and state vector $ x_{(t)} = \left \{ \begin{array}{c c} q_{(t)} \\ \dot{q_{(t)}} \end{array} \right \}$ ...
0
votes
2answers
64 views

My equations are inconsistent. Could someone help me see the error of my ways?

I'm given the following problem: Now, the following is my attempt at a solution: I have two problems: (1) With my equation for $v$, I end up having to take the log of $0$, which is obviously ...
0
votes
4answers
119 views

Newtonian Mechanics - Differential equation

If we combine Newton's second law of motion i.e. $F=m\ddot{x}$ and Newton's law of gravity i.e., $$ F=G\frac{mM}{x^2}, $$ where $x$ is distance, we obtain the following equation: ...
0
votes
2answers
57 views

Inverse of $r sin(\omega t) + v t$?

I am wondering if there is an inverse for this function, $x(t)=r sin(\omega t) + v t$. The inverse function theorem suggests that an inverse for this function does exist, although it may have to be ...
0
votes
2answers
35 views

Linear Oscillator without Friction

I already figured out the majority of the solution to this problem but I just need help on the last part. The question is: Consider the linear oscillator without friction: $$m\frac{d^2x}{dt^2}=-kx$$ ...
0
votes
1answer
129 views

Differential Equation - Falling Projectile - Help getting started?

Here is the question I'm dealing with: A ball with mass 0.15 kg is thrown upward with initial velocity 20 m/s from the roof of a building 30 m high. There is a force due to air resistance of ...
0
votes
1answer
39 views

Determine the motion for all time

In the frame $F=[0,\hat{k}]$, a particle of mass $m$, whose trajectory $[0,\infty)\xrightarrow{\rm r}\mathbb{R}$ is $r=z\hat{k}$ moves in response to a force ...
1
vote
1answer
124 views

Solving for Asteroid Orbit with Respect to Time

I am trying to create a differential equation with which I am can numerically solve to plot the orbit of an asteroid around Jupiter so far I have assumed the mass of jupiter is 0.001 of the mass of ...
0
votes
1answer
35 views

Temperature Diffusion, Laplacian

Find the temperature field, given that the temperature satisfies $\nabla^2 T = 0$, and T is a function only of r (working in cylindrical polars) I'm assuming I'm supposed to solve the Laplacian. So ...
1
vote
0answers
42 views

Particle in a Polya Vector field

For a given analytic function $H$ from $\mathbb{C}$ to $\mathbb{C}$, we define the Polya Vector Field to be $\bar{H}$. This then corresponds to a irrotational, conservative vector field on ...
2
votes
2answers
131 views

Centre of mass moves with constant velocity

The centre of mass of the Newton $n$-body problem is given by $$S=\frac{1}{M} \sum m_ix_i$$ with $M=\sum m_i$. Show that it moves with contant speed and hence has no acceleration. I don't understand ...
0
votes
1answer
100 views

Solution of pendulum linked to Weierstrass $\wp$-function

I've been working through a question about the equation of motion of a pendulum. I have to now solve the equation of the form: $$u'^2=u^3+au+b,$$ where $a=(\frac{g^2}{l^2}-\frac{c^2}{3})$ and ...
2
votes
0answers
439 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
0
votes
0answers
73 views

Close to giving up on this differential equation! Please help! [duplicate]

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
3
votes
2answers
122 views

pressure in earth's atmosphere as a function of height above sea level

While I was studying the measurements of pressure at earth's atmosphere,I found the barometric formula which is more complex equation ($P'=Pe^{-mgh/kT}$) than what I used so far ($p=h\rho g$). So I ...
1
vote
1answer
136 views

Method of Averaging an ODE

A few weeks ago, I was asked the following in a homework assignment Study the system $\dot{x}(t)=-\epsilon x(t)\cos(t)$ by the method of averaging and compare this to the exact solution. My exact ...
1
vote
1answer
178 views

Pursuit curves and arc length question

I am studying pursuit curves where a fast pirate ship which pursues a heavily laden treasure ship which tracks along a straight line. The ratio of the speeds of the ships is r > 1 (which is fixed) and ...
6
votes
3answers
308 views

two identical point charges can't collide

I've convinced myself intuitively that if you place two massless classical particles with the same charge in $\mathbb{R}^n$, with arbitrary initial velocities and (distinct) positions, they will never ...
1
vote
1answer
113 views

Phase space area preservation

What is wrong with the following argument? Suppose the initial configuration $(x,p)$ of a system of many non-interacting particles each of mass $m$ in phase space is given by a rectangle ...
2
votes
4answers
162 views

Three ball-spring system

So here is a crazy problem for you all. Imagine there is a system of three balls in a line. The first and last balls have a larger mass M and the middle ball is a smaller mass m. Inbetwen the two ...
3
votes
0answers
246 views

system of implicit nonlinear differential equations

Here I have a system of nonlinear differential equations: $ (M+2m)\ddot{x} + m(l_1 \ddot{\theta}_1\cos\theta_1 - l_1\dot{\theta}_1^2\sin\theta_1) + ...
2
votes
2answers
338 views

A differential equation of Buckling Rod.

I tried to solve a differential equation, but unfortunately got stuck at some point. The problem is to solve the differentail equation of hard clamped on both ends rod. And the force compresses the ...
1
vote
0answers
108 views

Collision of eigenvalues of a linear ODE (Krein collisions)

I am trying to understand the so called Krein collisions in Hamiltonian mechanics but I shall formulate the question in a rather general way. Suppose we have the following linear ODE: $ \dot{v}= ...
0
votes
1answer
213 views

A more elegant way of finding $\dot \phi$

Is there a simple way of obtaining an expression for $\dot{\phi}$ from the equation $$r{d^2\over dt^2}\begin{pmatrix} r\cos \phi\\r\sin \phi\end{pmatrix}=F'(r){d\over dt}\begin{pmatrix} r\cos ...
54
votes
3answers
68k views

Teenager solves Newton dynamics problem - where is the paper?

From Ottawa Citizen (and all over, really): An Indian-born teenager has won a research award for solving a mathematical problem first posed by Sir Isaac Newton more than 300 years ago that has ...
23
votes
1answer
2k views

Why isn't the 3 body problem solvable?

I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of ...
2
votes
3answers
613 views

Show velocity of a particle during its flight at time $t$

I'm completely stuck, I think I have to use Newton's second law but I have no idea where to start, any help would be appreciated! At time $t=0$ a particle of unit mass is projected vertically upward ...
3
votes
3answers
630 views

What exactly is meant by 'Integrate the Equation of Motion'?

Short Question: if a question says to 'integrate the equation of motion', what does it mean? Long Question: Question: Take a planet of mass $M$ and place a satellite at rest at a distance $R$ ...
1
vote
1answer
304 views

How can I put the “3 body problem” mathematically?

I'm trying to put the 3 body problem mathematically. But I don't know how. I always get something reasonable, but I get something that is wrong.
5
votes
0answers
1k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
2
votes
4answers
152 views

The motion of a system as a level set of the energy

Suppose we have a mechanical system with 1 degree of freedom, i.e. an ODE $$(1)\quad \ddot{q}+V^\prime(q)=0, $$ where $V \colon \mathbb{R} \to \mathbb{R}$ is some smooth function (potential ...
3
votes
1answer
623 views

Why does acceleration = $v\frac{dv}{dx}$

If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$ However I also understand $a=v \frac{dv}{dx}$. Can ...