Questions relating to Newton's Laws of Motion

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2
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1answer
50 views

motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
2
votes
1answer
43 views

Doubling Map and Measure

First off! This is a homework question, so I DO NOT want an answer to the question I'm writing, I really just want an explanation of the final bit (which I'll make clear). So if we have $T:[0,1)\to ...
2
votes
1answer
89 views

Confusion over notation in a book on the mathematics of QFT by Faria-Melo

While formulating this question, I arrived at a likely interpretation provided in an answer to my own question below. My problem appears to be one of inexperience in working with ambient coordinates, ...
1
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1answer
39 views

Elements in stiffness matrix

I do not know if I am at the right address here, but I'll just ask. Is the following correct? Every element in the stiffness matrix represents the displacement of every element, when exerting an ...
1
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1answer
139 views

Velocity vector from inertial to body frame

here's my question: I have position and velocity vectors of a body in the inertial frame. Now i need to switch the reference system to body frame. So i have $\bar{x}_b = \hat{R}\bar{x}_I$ where ...
0
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1answer
20 views

How do you get a measure space out of a dynamical system?

I'm reading a book on ergodic theory (by Cesar Silva), and also have read Stein and Shakarchi's third book on undergraduate analysis, where there is a section devoted to some ergodic theory. Both ...
0
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1answer
12 views

When finding the frequencies of normal modes, can you have a negative frequency?

Do you simply just consider the positive solutions? I tried a google search but didn't find anything quickly. The work I am studying is Lagrangian systems.
6
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0answers
250 views

Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...
5
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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 ...
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0answers
94 views

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 ...
3
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0answers
63 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 ...
3
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0answers
225 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) + ...
3
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0answers
70 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 ...
2
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0answers
55 views

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.
2
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0answers
398 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 ...
2
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0answers
71 views

Resolving mass of holy disk with moment of inertia?

A uniform lamina of mass m is bounded by concentric circles with centre O and radii a and 2a. the lamina is free to rotate about a fixed smooth horizontal axis T which is tangential to the outer ...
2
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0answers
460 views

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 ...
2
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0answers
79 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 ...
2
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0answers
167 views

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 ...
2
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0answers
72 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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0answers
25 views

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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0answers
33 views

Finding the centre of mass and moment of inertia about a point of a laminate

A triangular laminate OAB of uniform area density,$\rho$, has vertices at (0,0), (0,2) and (1,0). The moment of inertia about an axis through O,perpendicular to the plane of the laminate is denoted by ...
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0answers
36 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 \}$ ...
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0answers
40 views

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 ...
1
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0answers
28 views

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 ...
1
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0answers
33 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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0answers
50 views

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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0answers
69 views

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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0answers
81 views

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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0answers
35 views

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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0answers
36 views

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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0answers
65 views

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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0answers
125 views

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 ...
1
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0answers
91 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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0answers
180 views

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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0answers
130 views

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), ...
0
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0answers
51 views

fluid dynamics: sketching streamlines of velocity field when there is only one non-zero velocity component

I have been asked to sketch the streamlines in the $x_2$$x_2$-plane for the two-dimensional field: $$v=(x_1x_2,0,0)$$ All the examples I have seen of this kind of question use the $v_1$ and $v_2$ ...
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0answers
38 views

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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0answers
21 views

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 ...
0
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0answers
29 views

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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0answers
26 views

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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0answers
39 views

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 ...
0
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0answers
162 views

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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0answers
19 views

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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0answers
25 views

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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0answers
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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 ...
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0answers
38 views

Completely integrable geodesic flows without any degenerate point

Are there many examples of completely integrable geodesic flows (in the sense of Liouville), with say n integrals $f_1,\cdots, f_n$ such that everywhere, the differentials $(df_1,\cdots,df_n)$ are ...
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0answers
58 views

Bertrands theorem. The principle of least curvature.

Theorem: If a given set of impulses is applied to different points of a system in motion (either holonomic or nonholonomic), then the kinetic energy of the resulting motion is greater than the ...
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0answers
62 views

Modify formula for bouncing object to increase restitution

I originally started a thread over on Stack Overflow about this but it's diverged into mathematics which is way beyond my understanding. Basically I have the following formula (it's JavaScript but I ...
0
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0answers
431 views

Inertia tensor transformation under coordinate change

Let $I(x)$ be an inertia tensor in matrix notation of a body in a coordinate system $x\in R^n$. Under a coordinate change $x=\phi(y)$, does the tensor transform as $Dx^TI(\phi(y))Dx$, where ...