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Questions tagged [classical-mechanics]

For questions on classical mechanics from a mathematical standpoint. This tag should not be the sole tag on a question.

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What is the matrix of inertia of a thin rectangular plate? [on hold]

I only want to get sure how the matrix looks like, Is it : $\begin{bmatrix} \frac{Mb^2}{3} &0 &0 \\ 0&\frac{Ma^2}{3} & 0\\ 0& 0 & \frac{M(a^2+b^2)}{3} \end{bmatrix}$ Or ...
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35 views

Liouville measure in the energy level

Again banged my head trying to make sense of the Mechanics course and hit a rock bottom with another problem. I would really appreciate the step by step explanation for the results of it. We have a ...
2
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1answer
28 views

Brachistochrone step from Advanced classical mechanics (Bagchi)

I'm not sure as to how the book got from: $dx=\frac{a+b}{2}\int{\sqrt{\frac{1+\cos(\theta)}{1-\cos(\theta)}}\sin(\theta)d\theta}$ to: $x=\frac{a+b}{2}(\theta-\sin(\theta))+constant$ where $a$ and $...
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1answer
33 views

Canonical transformations on cotangent bundle

We got as part of homework for Mechanics this exercise. As the course was a little chaotic I barely got grip of some notions and I feel a bit lost so any solution would be welcomed. Let V be a ...
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14 views

Why is translational energy part of the equation for kinetic energy for the first leg of a double pendulum?

Isn't all kinetic energy of the first leg already in the rotational energy based on the angular velocity theta_1_dot? https://en.wikipedia.org/wiki/Double_pendulum#Lagrangian (i'm referring to 1/2*...
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1answer
71 views

Why is symplectic geometry the natural setting for classical mechanics?

I was reading this very nice document, to understand why symplectic geometry is the natural setting for classical mechanics. I more or less understood why there is naturally a 2-form that arises. ...
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10 views

rigid body inertia tensor matrix

We now consider a disk of radius R (modelling a wheel) rolling down an inclined plane with inclination α. We assume that it is rolling without sliding. To describe this motion we use the following ...
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1answer
34 views

Mechanics - Banked Slope

I have an example in a textbook that goes like this: A car travels around a bend which has radius 100 m and is banked at an angle of 20° the horizontal. The car is travelling at a speed of 30 ms-. ...
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21 views

particle attached to a string and projected

I'm stuck in the following question: One end of a light inextensible string of length $a$ is attacheded to a fixed point $A$, a distance $\frac{13a}{27}$ below a horizontal ceiling. The other end ...
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1answer
36 views

Two dimensional motion with variable acceleration

I'm doing a question in a textbook which is known to sometimes have wrong answers. However, it's more likely that I'm just being stupid. Particles P and Q, each of mass 0.5 kg, move on a horizontal ...
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2answers
48 views

Writing 2D linear system of balance laws in compact form

I have three equations $$\rho \frac{\partial }{\partial t}v = \frac{\partial}{\partial x}\sigma_{21} + \frac{\partial}{\partial z}\sigma_{23}$$ $$\frac{\partial}{\partial t}\sigma_{21} = a\frac{\...
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Jet bundle cohomology

Consider a base manifold $\mathcal{M}$ and a smooth bundle $E\to\mathcal{M}$. I am interested in the cohomology groups of the variational bicomplex associated with the jet bundle $J^{\infty}E$. In ...
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1answer
13 views

How to model this problem as a boundary value problem of PDE system

I am trying to solve this system of elasticity partial differential equations in 2D in cylindrical coordinates, where the model is axysimmetric, around the $z$ axis. $$\frac{1}{r} \frac{\partial (r\...
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0answers
8 views

condition for existence of phase flows 3

first I apologize for my bad English level. hope somebody understand the question. in "mathematical methods of classical mechanics" by Arnold, is a problem which: prove that a system with potential U=-...
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24 views

Classical Rigid body Mechanics

A thin heavy disc can turn freely, about an axis in its own plane, and this revolves horizontally with a uniform angular velocity $\omega$ about a fixed point on itself. Show that the inclination $\...
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20 views

unsure how to expand a LAX pair for KDV equation

This is my first post so im not sure how to make it all mathsy so im going to write it on here, I know that to find the lax equation you find [LM-ML]=0 but im struggling to follow the expansion, for ...
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1answer
35 views

The trajectory of the harmonic oscillator

If we consider the Hamiltonian for the simple harmonic oscillator given by, $$H(p,x) = \frac{p^2}{2m}+\frac{kx^2}{2}$$ where $m$ is the mass, $k$ is the stiffness and $p$ is the momentum, then the ...
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0answers
37 views

Lagrangian density of second order hyperbolic equation

It's been some hours now that I am trying to find the Lagrangian density of the following hyperbolic PDE with variable coefficients and $c_{ij}=c_{ji}(x)$ $$ \partial^2_t u - c_{ij}\partial_i\...
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1answer
36 views

Systematic way of obtaining conservation laws in dynamical systems

Motivation Consider a point particle of mass $m$ moving in $\mathbb{R}^3$ under the influence of some force field $\vec{F}(\vec{r},t)$. The fundamental equation governing the dynamics of this system ...
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26 views

What is proof of direction of Change in position vector $d \vec r$

How can I prove that the direction of an infinitesimal change in position vector $\mathrm d\vec r$ is the same as that of the instantaneous velocity $ \vec v=\mathrm{d}\vec r/\mathrm{d}t$? What I ...
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18 views

how to prove that the radius of curvature is equal to the radius of the circle?

A material point M describes a circle of radius R and the velocity vo, k (v, a) = alpha - constant, whatever> = t0. It is shown that the radius of curvature is equal to the radius of the circle
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1answer
43 views

Solving a differential equation for oscillatory mechanics

Well I was studying oscillatory mechanics and I got stuck on an differential equation which I am not able to solve. The equation is $$ ma= ( -kx) + ( -bv) + F\sin(¥t+\Delta) $$ where $a$ is the ...
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Formula for crank/piston setup

I’m not totally certain how to ask this question or what mechanism this is, but I’m trying to know how far the bottom of a vertical object will travel when pushed straight down from the top. ...
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1answer
37 views

How to determine the direction of tension?

I'm having trouble determining the direction of tension confidently for harder examples. For example, A light, smooth ring, $R$, is threaded on a light, inextensible string. One end of the string ...
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2answers
38 views

Elastic collision between a circle and a point

In a 2D environment, I have a circle with velocity v, a stationary point (infinite mass), and I am trying to calculate the velocity of the circle after a perfectly elastic collision with the point. ...
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87 views

Are Hamiltonians proper maps?

In the context of classical mechanics: Which assumption on a Hamiltonian function $H : T^* \mathcal M \to \mathbb R$ is most reasonable to assure its phase flow $\varphi_t : T^* \mathcal M \to T^* \...
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1answer
66 views

Lagrangian mechanics- conservation of energy

Consider a single particle system whose Lagrangian remains the same if the position of the particle is simultaneously (i) rotated by an arbitrary angle s about the z-axis and (ii) shifted by an amount ...
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23 views

Reference request on an equation while studying Stokes steam-functions

I came across this equation while studying Stokes steam-functions and I'm not sure how its derived. The equation is $$(r^2-\frac{a^3}{r})\sin^2(\theta)=b^2$$ I believe it is the equation of a ...
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1answer
38 views

Applying a linear transformation to a system of differential equations

I am reading the book: Nonlinear Oscillations, Dynamical System and Bifurcations of Vector Fields (Guckenheimer and Holmes), chapter 2: An introduction to chaos. About Van der Pol's equation, it can ...
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1answer
45 views

Converting Cartesian Components of a Vector on a Sphere to Tangential-Normal Components

I am dealing with a mechanics problem on a unit sphere where all the equations are written in terms of Cartesian components and wish to make a conversion to a system $n, t, s$ where $n$ is the unit ...
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1answer
27 views

The expression “independent” in the context of Analytical Mechanics

In Analytical Mechanics, as physicist, we often use the expression $q_1, q_2$ are independent coordinates; however, this is a different concept than what we, as mathematicians, call linearly-...
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1answer
36 views

Where is it used that the symmetry conserves the symplectic form in noether's theorem.

For the proof of Noether's theorem, it seems like that the only thing that's important for the symmetry map $S_g : M \to M$ is that it conserves the Hamiltonian (which will then imply that the moment ...
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39 views

What is symmetry in physics in the mathematical sense, using (Lie) groups

In physics, we sometimes say that, for example, a certain classical system has a certain symmetry, which is given by some group. I don't feel like I understand this well enough. Are there some good ...
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Different kinds of stability that apply to planar periodic orbits, and what do they mean?

This is a question about terminology related to orbit stability. I had wanted to ask about stability of orbits described in the paper Three Classes of Newtonian Three-Body Planar Periodic Orbits ...
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2answers
21 views

Complex SUVAT help?

Two objects are dropped from the top of a cliff height $H.$ The second is dropped when the first has travelled a distance $D.$ Prove that the instant when the first object has reached bottom, the ...
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11 views

Demostration of the minimun radius of curvature

I have a doubt about how to demostrate that the minimun radius of curvature is when the particle is at the maximum point. Using this formula:$\dfrac{\left[1+\left(\dfrac{\operatorname d \!y}{\...
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0answers
14 views

Biconformal space and curvature

I've found very few contributions about the so called Biconformal Space, "a curved phase space". I was sure that in general phase spaces are cotangent bundles naturally equipped with a symplectic 2-...
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1answer
18 views

Understanding the forces on a simple model

A light elastic string AB has natural length $1.25m$ and modulus of elasticity $24.5N$. Another light string CD has natural length $1.25m$ and modulus of elasticity $26.95N$. The two strings AB and ...
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23 views

How to maintain constant line length around a rotating joint

I'm working on a design for a robotic leg that uses strings as "tendons" for extending and retracting the leg. This is as much a geometry problem as it is a mechanical engineering problem. Refer to ...
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0answers
42 views

Change of coordinates and effects on the tangent bundle of a Manifold

I am doing ex. 2.3(2) from Frankel's book "The geometry of Physics". He says to consider the tangent bundle to a manifold M and show: i) that under a change of coordinated in M, $\partial / \partial ...
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64 views

Understanding the notation when finding action-angle coordinates

I'm trying to learn the basics of KAM theory and I wanted to first get to grips with Liouville integrability for Hamiltonian systems but I'm getting rather confused by the notation which seems to be ...
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0answers
29 views

classical mechanics in coadjoint orbits

We know that coadjoint orbits are symplectic manifolds, and they can be used to find unitary representations of lie groups and stuff, and it's also related to quantization. However, is it true that ...
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1answer
33 views

Word problem regarding finding torque of a shoulder

Question: What is the torque about the shoulder if the arm is held in an abducted position at 60 degrees from the body in the frontal plane while holding a 10 kg dumbbell? Assume that the mass of the ...
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0answers
53 views

Energy Conservation and Speed of a Falling Satellite

I've been given the following problem (admittedly on a homework sheet), which I've solved, but I feel there has to be a neater method of solving it: A satellite falls freely towards the Earth ...
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0answers
34 views

Examples of prime ideals for Lie algebras

I am looking for examples of prime ideals for Lie algebras. In particular, I am interested in examples involving the Lie algebra given by the commutator of endomorphisms of a complex vector space and ...
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2answers
173 views

The physical meaning of a symplectic form.

So I've studied a bit about symplectic geometry, and I know that phase space is a symplectic manifold, and the symplectic form induces a poisson bracket. However, what is the physical meaning of the ...
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Minimising a complex equation

I'm trying to find a way to find a value(s) of $\omega$ which minimise $$\mathbb{R} = 1 - \frac{1+R_0^4-2R_0^2}{1+R_0^4 -2R_0^2\cos(2\omega\beta_1 L)}.$$ The terms in the equation depend on $\omega$,...
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1answer
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Eulerian to Lagrangian Coordinates Help: How to find constants? (Theoretical Mechanics)

Consider material traveling in a two-dimensional space with its velocity at any given location in space (capital letters correspond to Eulerian, lower case to Lagrangian) given by $\...
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2answers
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Mechanical vibration: single degree of freedom model of wheel mounted on a spring

I think the author of my book has this solution wrong and would like some feedback on my thoughts (see Example 1.4.1). The solution states that the total energy of the system is represented by the ...
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24 views

Inverting the WZW Lagrangian to its Poisson bracket

Could someone give me a hint or how one should start for inverting the WZW Lagrangian or equivalently the Kirillov-Kostant 2-form: $$\Omega = \frac{K}{4\pi} \int_0^{2\pi} \mathrm d x \text{ Tr} (g^{-1}...