Questions tagged [classical-mechanics]

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

Wikipedia says:

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Besides this, many specializations within the subject deal with gases, liquids, solids, and other specific sub-topics. Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light.

For questions on classical mechanics from a mathematical standpoint. This tag should not be the sole tag on a question. Examples of other tags that might accompany this include , , and .

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Teenager solves Newton dynamics problem - where is the paper?

From Ottawa Citizen (and all over, really): An Indian-born teenager has won a research award for solving a mathematical problem first posed by Sir Isaac Newton more than 300 years ago that has baffled mathematicians ever since. The solution…
jnm2
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Why isn't the 3 body problem solvable?

I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of freedom, then the system is solvable in terms of…
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Tensors in the context of engineering mechanics: can they be explained in an intuitive way?

I've spent a few weeks scouring the internet for a an explanation of tensors in the context of engineering mechanics. You know, the ones every engineering student know and love (stress, strain, etc.). But I cannot find any explanations of tensors…
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Conservative Systems and The Hamilton-Jacobi Equation

I am trying to understand geometrically the relation between a conservative classical system described by the hamiltonian $H$ for which the trajectories of particles are given by $$\dot{x} = \nabla_p H(x,p) \qquad \dot{p} = - \nabla_x H(x,p)$$ and…
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What is minimum speed needed to jump over sphere object that has radius R and at distance d?

(I am not expert in English. I will write as well as I can.) To understand this question easier, lets see this picture. From this picture, what is minimum initial speed that this grasshopper need to jump over this log? The grasshopper movement path…
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Alternatives to the classical pendulum ODE

The classical (undamped) pendulum ODE is $$\ddot \theta = -\frac{g}{\ell} \sin(\theta)$$ Defining state ${\bf x} := (\theta, \dot \theta)$, we have a system of $2$ ODEs, $\dot {\bf x} = {\bf f} ({\bf x})$, where ${\bf f} : \Bbb R^2 \to \Bbb…
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Why don't we differentiate velocity wrt position in the Lagrangian?

In Analytic Mechanics, the Lagrangian is taken to be a function of $x$ and $\dot{x}$, where $x$ stands for position and is a function of time and $\dot{x}$ is its derivative wrt time. To set my question, lets consider motion of a particle along a…
17
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How to introduce stress tensor on manifolds?

I want to understand the type of stress tensor $\mathbf{P}$ in classical physics. Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text d \boldsymbol s$ (vector) equals $\text d…
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Arnold's theorem on action-angles.

I changed the question slightly in its form to make it more readable. I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question) If you…
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What is symplectic geometry?

EDIT: Much thanks for answers. As was pointed out, the question as it stands is a little too broad. Nevertheless, I don't want to delete it, because I think that such introduction-style questions can be answered without writing a book, rather…
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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 of solving the effect of three bodies which all…
14
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Coordinate-free techniques in Lagrangian mechanics

Consider the following Lagrangian (Exercise 3.6B from Abraham and Marsden's Foundations of Mechanics): $$ L(\upsilon)=\frac12g(\upsilon,\upsilon)+V(\tau_Q\upsilon)+g(\upsilon,Y(\tau_Q\upsilon)) $$ ($V \colon Q \rightarrow \mathbb{R}$ is a smooth…
akater
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Publication date of the book of 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 love it. I am really looking towards the sequel(s)…
JoseDS
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How to solve harmonic oscillator differential equation: $\dfrac{d^2x}{dt^2} + \dfrac{kx}{m} = 0$

I am able to solve simple differential equations like : $$\dfrac{dy}{dx} = 3x^2 + 2x$$ We simply bring $dx$ to other site and integrate. But how do we find solutions of differential equations like : $$\frac{d^2x}{dt^2} + \dfrac{kx}{m} = 0$$ ? We…
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Envelope of Projectile Trajectories

For a given launch velocity $v$ and launch angle $\theta$, the trajectory of a projectile may be described by the standard formula $$y=x\tan\theta-\frac {gx^2}{2v^2}\sec^2\theta$$ For different values of $\theta$ what is the envelope of the…
Hypergeometricx
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