Questions relating to Newton's Laws of Motion

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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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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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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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64 views

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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69 views

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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263 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) + ...
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73 views

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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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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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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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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86 views

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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76 views

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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Bullet hit, calculate pressure

I wanna calculate the stress that is there when a bullet hits a metall plate. i know the formula pressure = force / area But how can I calculate the force?
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Matrix linearization of the Lagrangian points.

I have to solve a long problem, and I´m in trouble in a step. The step is to linearize the next differential equation, by writtin its correspondient Jacobian, and then, calculate the eigenvalues of ...
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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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52 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 ...
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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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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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118 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}= ...
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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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Lax Pairs and constant eigenvalues

Can someone tell me whether the following is true, and if so a hint the proof? If we have a Lax Pair $\dot{L} = [A,L]$ then the eigenvalues of $L$ are constants of the motion. (The opposite ...
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representation of Eulers's equation in biharmonic form

As we know the Euler's equation $${\rm div}{\rm div}(\frac{\nabla^2F}{\|\nabla^2F\|})=0$$ Can be written in biharmonic equation form $$\Delta^2F+ (something)=0$$ I want to know in the context of solid ...
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Balance of forces in a mechanics problem

I tried to solve a particular problem of mechanics and found some difficulties in the vector analysis part that I can't get rid of. It's probably some stupid mistake I made, but I can't see it now, ...
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At how much horizontal distance the body will strike the ground?

A very smooth plane of length 40m is inclined to horizon at an angle 30 degree . From the foot of this plain , a body starts from rest and moves with an acceleration $10m/s^2$ . After moving a ...
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Showing that the vivial theorem holds for the case of a particle in a closed orbit with (f = -kr) in terms of E

I'm trying to show that the vivial theorem holds by explicitly calculating the time average of potential in terms of the total energy E and showing they're the same computed both ways.This is for a ...
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Show that after the deformation the line element is given by the following equation

Assume throughout that $X_i,x_i$ (i=1,2,3) are respectively the material and spatial coordinates of a point referred to a common rectangular Cartesian coordinate system with origin $0$, and the the ...
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Using Feynman's Subscript Notation

I have a homework problem that wants me to calculate the force $\vec{F} = \vec{\nabla}_{\vec{X}}U + \frac{\mathrm{d}}{\mathrm{d} t} \left(\vec{\nabla}_{\dot{X}} U\right)$ where $U(\vec{X}, \dot{X}, ...
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Lagrangian of particle on a horizontal, square rotating hoop.

The problem of interest is worded as follows: A horizontal square wire frame with vertices $ABCD$ and side length $2a$ rotates with constant angular frequency $\omega$ about a vertical axis through ...
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Estimate the period of motion of a particle

a) Evaluate the period of motion of a particle in the potential in the field $U(x)$ if it's energy value lies in the vicinity of $U_m$ (see the picture below). b) Estimate at what part of of period ...
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What are sliding vectors mathematically?

What is the mathematical definition of sliding vectors and their operations, as used in mechanics? What kind of mathematical structure do they form? Does the operation of constructing the "space" of ...
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Deriving the $F_3$ type generating function in Hamiltonian formulation

I'm working on some practice questions and I am a bit confused with this one: Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition ...
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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