Questions relating to Newton's Laws of Motion

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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 ...
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Angular momentum cylindrical coördinates

From "Classical Mechanics" - Taylor, problem 3.30 Consider a rigid body rotating with angular velocity $\omega$ about a fixed axix. (You could think of a door rotating about the axis defined by ...
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The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
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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 ...
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Hyperbolic Motion in a Central Field

I have to give a 30 mins lecture this coming Thursday in my classical mechanics class (graduate level in math department, with Arnold as the primary text) and I am really struggling to find any good ...
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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) + ...
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what is the domain of the Lagrangian of a surface embedding?

If we view our Lagrangian particle mechanics geometrically, then we describe a particle trajectory as a map from R to a manifold, and the Lagrangian $L(x,\dot{x})$ as a function on the tangent bundle ...
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“Straightforward” application of trigonometry to the slingshot effect/gravity assist

I have been trying to understand the formula $$v_f^{2}=v_i^{2}+2V(V(1-cosβ)+v_i(cos(α-β)-cosα))$$ as it relates to Fig. 2 on page 5 of this exposition: ...
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Holonomic constraints and degrees of freedom

Back in my undergrad I learned that in a dynamical system, if I add a holonomic constraint, I subtract one degree of freedom from the space of configurations. But one can think of situations in which ...
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A Proof of Bruns' Theorem

I am looking for a proof of Bruns' Theorem. Theorem: The 10 classical integrals of the three-body problem are the only algebraically independent integrals of this 18-degree-of-freedom system.
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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 ...
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Publication date for Michael Spivak - Physics for Mathematicians II?

I bought the book "Physics for Mathematicians I" by Michael Spivak (http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322), have worked through quite some chapters and ...
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Sign problem with Poisson brackets

I am wondering if anyone could explain to me either why my method is not valid or point out where I have made an algebraic slip. I have been looking at this for a long time, to no avail. Let $\{\cdot ...
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Motion on a parametric surface

Please excuse what will surely turn into a long rambling question, full of incorrect terminology. I'm trying to figure out the mathematics of moving on a parametric surface - that is, for some ...
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A lower positive bound on the number of closed orbits with given energy for a mechanical system

Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the ...
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Books on Mechanics

I was wondering if any of you know the names of some good books that give an introduction to langrangian and hamiltonian mechanics. I've finished kleppner and kolenkows introduction to mechanics and ...
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Speed of a horizontal rotating lamina after a transfer of momentum from a body attached by a string

Four uniform rods, each of mass $m$ and length $2l$, are joined rigidly together to form a square frame $ABCD$ of side $2l$. The frame is placed with all four sides at rest on a smooth horizontal ...
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A problem related to Vectors.

A few days ago I posted an answer to a question on Phys.SE. The question is: Three particles $A,B$ and $C$ are at the vertices of an equilateral trinagle $ABC$. Each of the particle moves with ...
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Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
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Tensorial product, a simple question

I need to find the components of $D$: $$D=a\otimes a$$ where $a$ is a tensor of order 2. Thanks!
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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 \}$ ...
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Derivation of Euler lagrange for Yang Mills

I need someone to sketch the conventional steps(from variation to vanishing of arbitrary function chosen , etc, etc) of Classical Yang-Mills. If using exterior product product could you emphasize any ...
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Horizontal drift of snowflake

I wonder if the random-walk dynamics of falling snowflakes is understood well enough to estimate the likely sideways drift of a single snowflake falling in a windless environment, from its cloud ...
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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 ...
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The equvalence of the virtual work and the Hamiltonian equations

I am reading Whittaker's Analytical Dynamics. This is chapter 10 *Hamiltonian Systems&. Paragraph 109 is Hamiltonian Systems & Their integral invariants. Whittaker starts with the Lagrangian ...
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What is $\int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \; $?

Given $F[u]$ and $G[v]$ are functionals of a real-valued function, what is $$ \int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \quad ? $$ I have encountered such expressions for ...
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Verify that $r=a(1-e\cdot \cos(\theta))^{-1}$ is a solution of the central force equations

A particle of mass $m$ moves under the influence of an attractive central force of magnitude $mk/r^2$ where $r$ is the distance from the origin. I have the equations $$ r^2\cdot \frac{d\theta}{dt} ...
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period of motion on the phase curve

I am interested in the following question. (It is a rephrased problem in Arnold's book "Mathematics methods of classical mechanics" (2nd ed. page 20)). Given are potential function $U(x)$ such that ...
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Regularization theory

In order to remove the collision singularity in the equations of motion of the three dimensional two body problem, one defines the coordinate transformation $x_1=u_1^2-u_2^2-u_3^2+u_4^2$ $x_2=2(u_1 ...
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Mechanics with elastic strings

Two elastic strings, A and B, stretch by 30mm and 60mm respectively when a weight of 4N is attached to each in turn. the strings are hung vertically from the same point, close together, so that when ...
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Applications of mathematics to some kinds of sporting strategies

I am a rather newbie maths person. Haven't studied maths in a while and so not sure what things are called was hoping to get some information to push me in the right direction so I know what it is I ...
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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}= ...
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Closed Geodesics as minimisers of action functional

Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
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Transformation of the dynamics of mechanical system under coordinate change

It is well known that the dynamics equation for a mechanical system (e.g. a robotics manipulator) is given be the Euler-Lagrange equation which takes the particular form (in the simplified version), ...
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Newton's method for the brachistochrone

Consider the potential $V(x,y)=-y$ and a particle at rest in the beginning of the coordinate system. We are going to examine the brachistochrone - the smooth curve of fastest descent. Assume we are ...
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Analyze rotation of satellite orbit due to transverse acceleration.

Consider a small satellite which moves in a 2D elliptical orbit around a much larger body (e.g. the Sun) under the influence of Newtonian gravitational acceleration $$Ar=G.M/d^2$$ QUESTION:- Is ...
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Curvilinear Coordinates and basis vectors

In the following notes, http://www.maths.manchester.ac.uk/~wparnell/MT20401/MT20401_lecture3.pdf $\frac{\partial \vec{r}} {\partial q_i}$ forms a basis set for the vector space. How does this happen? ...
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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 $$ ...
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How to evaluate a velocity dependent force integral?

Basically, this is a question from Classical Mechanics by John Taylor. I know how this problem goes but my problem is how to evaluate this integral in order to find v as a function of time?
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Classical Mechanics by John R. Taylor Chapter 1 Problem 40

This is a problem from Classical Mechanics by John R. Taylor Chapter 1 Problem 40 I got an answer of 70.5deg but I'm not sure if it is correct
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Trajectory question

With initial conditions $ x(0)= l > 0 $ and $\dot{x}(0)=0$ ,determine the trajectory $x(t)$ of the particle. Also determine its position and velocity at times $$\frac {1}{2}T,\frac {1}{3}T,\frac ...
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Finding angular momentum about the center of mass?

If we have a couple of particles of an equal, unknown mass: $r_{+} = (c + e^{-Bt} \cos({\theta}))\textbf{x} + (d + e^{-Bt} \sin({\theta}))\textbf{y}$ $r_{-} = (c - e^{-Bt} \cos({\theta}))\textbf{x} ...
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Lagrangian of a pendulum inside af disc

I'm currently struggling with a mechanical problem, where I need to find a relationship between the two angles in a mechanical system. The two equations of motions were derived using the Lagrange ...
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continuum mechanics: deriving eulerian 'conservation of mass' and 'balance of linear momentum' equations for cylinder with variable cross-section

I know the equations for conservation of mass and balance of linear momentum for a 1 dimensional cylinder with fixed cross-sectional area, A. Is there a way of deriving these equations in fluid ...
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Simple mechanics question - trapeze

A woman of mass 50kg swings on a light trapeze, i.e. a light seat suspended from a fixed point by a light rope. Her centre of mass, G, moves on the arc of a circle of radius 9m and centre ...
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Line Integral Problem best or easier solved using geometry?

Does anyone have any recommendation on a line integral problem involving vector fields (aka work) such that evaluating the resulting line integral using parameterization would be significantly ...
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Short proof of Jacobi identity for the Poisson bracket — is this valid?

I've been trying to make a short proof for the Jacobi identity for the Poisson bracket on phase space. My idea goes like this, we know the following: $$ \frac{\mathrm{d}}{\mathrm{dt}} \{f,g\} = ...
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Small oscillations, Lagrangian about equilibrium

I have a Lagrangian $$ L = \underbrace{\dot{x}^2 + \frac{1}{2}\dot{\theta}^2 + \dot{x}\dot{\theta}\cos\theta}_{\text{kinetic energy}} \underbrace{ - \frac{3}{4}x^2 + \cos\theta}_{\text{-ve potential ...
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Spinors and rotations in 3D

I have a cylinder with the axis coincident with the $x$ axis of a $3D$ euclidean reference frame. I have three possible rotations: 1) Roll: rotation of an angle $\Psi$ around the $x$ axis. 2) ...
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Linearising angle to chord length over a reasonable domain of < pi; or, how to make measuring a rock with a protractor easy

Imagine that I have a protractor and compass, and wish to use it to measure the distance between two points (potentially in three dimensional space, such as on a rock). However also being a forgetful ...