Questions relating to Newton's Laws of Motion

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When is a vector field hamiltonian with respect to some symplectic form?

Given a vector field $v$ on a $2n$-dimensional manifold, how many symplectic forms are there on $M$ that make $v$ a hamiltonian vector field? Alternatively, take the set of all $(H,\omega)$ pairs, mod ...
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458 views

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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871 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 ...
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40 views

rigid body poisson bracket

i have trouble to understand the definition of the rigid body poisson brackets. In the book of Marsden and Ratiu "Introduction to Mechanics and Symmetry", in chapter 10.1 they introduce the poisson ...
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37 views

Numerically iterating the dynamics of a constrained Newtonian system

This question is about the dynamics (in classical mechanics) of a rigidly linked chain of $N$ point masses, see figure. Let us say that the masses $m_1,\ldots,m_N$ have initial positions ...
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101 views

Axioms of Newtonian Mechanics

Axiomatically speaking, could Newton's laws be derived (as theorems) from the conservation of momentum and energy -- along with a few suitable definitions of things like an inertia frame and force? ...
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176 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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107 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 ...
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37 views

Have I gone wrong here? (partial differentiation)

Starting with $T = \frac{1}{2}M_{w}\dot{x}^{2} + \frac{1}{2}I_{w}\frac{\dot{x}^2}{r^2} + \frac{1}{2}M_{b}((\dot{x} + L\dot{\theta}cos(\theta))^2 + (L\dot{\theta}sin(\theta))^2) + ...
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75 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 ...
3
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107 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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80 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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301 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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78 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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32 views

Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
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22 views

Moser's Twist Theorem for maps with reflection

Suppose I have a simple two dimensional integrable twist map, such as $x_{1}=x_{0}+y_{0}, \quad y_{1}=y_{0}$. Suppose that I perturb it in such a way that Moser's Twist Theorem is satisfied. What ...
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260 views

Angular momentum in Cylindrical Coordinates

How to calculate the angular momentum of a particle in a cylindrical coordinates system $$x_1 = r \cos{\theta}$$ $$x_2 = r \sin{\theta}$$ $$x_3 = z$$ Thanks.
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158 views

“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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39 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 ...
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118 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.
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839 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 ...
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92 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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240 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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78 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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Simple Harmonic Motion; Tension in Elastic rope

I'm struggling to model this question out correctly. A glider and its pilot have total mass $230$ kg. The glider lands on a horizontal airstrip and when its speed is $16$ m/s it hooks on to the ...
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19 views

Pullback of a Hamiltonian

I understand that a Hamiltonian vector field $H$ creates a Hamiltonian flow $\phi_t$. Now, in order to prove that the Hamiltonian is conserved one uses the following \begin{eqnarray*} ...
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29 views

Everything about Legendre transform

The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
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26 views

Is time-1 map of a Hamiltonian vector field defined on a cylinder always twist?

Suppose I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are ...
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53 views

Finding equations of motion for inverted pendulum using Lagrange

I studied engineering 10 years ago but am struggling to remember how to find the equations of motion for this project of mine. I have a two wheeled inverted pendulum that I want to balance. $M_w = $ ...
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0answers
19 views

Relation of the additivity of Lagrangians of non-interacting mechanical systems to existence of a globally defined flow

Reading Landau's Mechanics, this question came in my mind. Suppose we have two non-interacting mechanical systems A and B, individually made up of a system of particles that are known to interact in ...
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27 views

Infinitesimal Transformation. Importance of Infinitesimal Transformation

My teacher ask me to start to study about Hamilton systems, Noether theorem. I am not advanced in that kind of study, so the teacher want from me to see some easy research, some proofs, step by step. ...
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64 views

Canonical Transformation and Symplectic Conditions

I have one question regarding Canonical transformation and symplectic matrix. I have read some notions from the following note: http://www.chim.unifi.it/orac/MAN/node6.html For me it is not clear ...
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Poincaré-Birkhoff theorem in sympl. geometry

On p. 274 of McDuff and Salamon's Introduction to symplectic topology a corollary to the Poincaré Birkhoff theorem is presented. So we are given an area preserving map on an annulus ...
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String problem with two equal particles

Two equal particles are connected by a string one point of which is fixed and the particles are describing circles of radii $a$ and $b$ about this point with the same angular velocity so that the ...
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0answers
29 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
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0answers
50 views

Scaling Two Equations

I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated. An incompressible thermal conducting fluid is contained between two infinite ...
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Trouble understanding Poisson Brackets

I'm looking at page 94 here - I understand the definition of Poisson brackets at the top of the page (which uses summation convention) but I don't get why the calculations in (4.61) are true. I'm ...
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29 views

logic verification angular momentum

So I have the following question: Your given a uniform right circular cone with a half angle at the apex of $\alpha$, a height of b and radius of $p_0$. Choose a coordinate system $O_{xyz}$ such that ...
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26 views

Show the relation $W$ is constant

If the space $W$ is constant (doesn't move with the flow), show that $$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV=-\int_{\partial{W}}\rho \left ...
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Visco-elastic fluid reference

What is a good book on visco-elastic fluids for self-teaching after one has studied Gurtin's Intro to Continuum mechanics? Thanks!
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Proving that a certain function is an integral of motion for a Hamiltonian

Let $H=q_1p_1-q_2p_2-aq_1^2+bq_2^2$ (with $a,b$ constant) be a Hamiltionian. Show that $G=\dfrac{p_1-aq_1}{q_2}$ is a first integral (integral of motion) of this system. According to the ...
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Forms and conservative forces

According to Tongs notes on Classical Mechanics; a force is called conservative when $F=-\nabla V$ And iff $\nabla \times F = 0$. This is in $R^3$. Also the potential $V=\int_{x_o}^{x^1} F(x)$ $dx$ ...
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104 views

How to solve an overconstrained system of equations?

What is the easiest codeable way to solve an overconstrained static model? How does Force Effect https://forceeffect.autodesk.com do it? Given a 10m long bar angled as the hypotenuse of a 3, 4, 5 ...
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33 views

Oscillation of a disc about a rod perpendicular to the disc but not through the centre.

A thin uniform circular disc of radius a and centre A, with density p, has a circular hole cut in it of radius b and centre B, where $AB = c < a−b$. The disc is free to oscillate in a vertical ...
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Mechanics - Kinematics in two dimensions, how to find the time given a quadratic?

I have this information: $$u = 2i + 3j$$ $$r0(\text{initial position}) = 40i + 20j$$ $$r = 52i + 128j$$ $$a = -0.06i -0.04j$$ I need to find the $t(\text{time})$ at this point. I can use the equation ...
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27 views

Mass density function verification in continuum mechanics

I'm having problems knowing where to go with this question: "Suppose that a certain material initially fills an open region $O \subseteq \mathbb{R}^3$, with $\Omega := [−1, 1]^3 \subset O$, so in ...
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116 views

Integration over a second order tensor

I would like to compute the mean value of a second order tensor $\mathbf{T}$ expressed in planar cylindrical coordinates. The mean value for any second order tensor is (reference [1] page 101) ...
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Find the value of P when the sliding begins

I was able to do the first part. For the second part, I wrote down $$F_b=\frac{5}{6}N_b$$Then taking moments about $C$ I got $$Pcos\theta +N_bsin\theta=F_bcos\theta$$ Then resolving horizontally and ...
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0answers
59 views

Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
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Invariants under Hamiltonian mechanics?

I am interested in certain properties of measures evolving according to Hamiltonian mechanics. Say we have a point $z$ in phase space: $z = (p,q)$ where $p$ is a generalized momentum vector and $q$ is ...