Classical mechanics refers to the classical (i.e., non-relativistic, non-quantum) study of physics. Three major formulations of classical mechanics are newtonian mechanics, lagrangian mechanics, and hamiltonian mechanics. The latter two are rather useful in extensions to Classical Mechanics; ...

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When is the principle of virtual work valid?

The principle of virtual work says that forces of constraint don't do net work under virtual displacements that are consistent with constraints. Goldstein says something I don't understand. He says ...
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How do traveling waves pass through a standing wave node, if the node doesn't move?

I'm having trouble with the explanation that a standing wave in a string is the superposition of traveling waves. The nodes in the diagram above are points where the particles of the string's ...
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Rotationally invariant body and principal axis

Suppose a rigid body is invariant under a rotation around an axis $\mathsf{A}$ by a given angle $0 \leq \alpha_0 < 2\pi$ (and also every multiple of $\alpha_0$). Is it true that in this case the ...
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868 views

Pendulum with water dripping out

Consider a pendulum, consisting of a string of length $l$ tied to a ball of negligible mass and radius $r$. The bob is filled with water, which has density $d$, and the pendulum is given a small push ...
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187 views

Naive questions on the concept of effective Lagrangian and equations of motion?

Let us consider a LC circuit containing an electric dipole moment, the quantum system (electric field $E$ coupled with a dipole moment) can be described by the path integral $$Z=\int DEDxe^{i\int ...
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Hamiltonian and the space-time structure

I'm reading Arnold's "Mathematical Methods of Classical Mechanics" but I failed to find rigorous development for the allowed forms of Hamiltonian. Space-time structure dictates the form of ...
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Can a car get better mileage driving over hills?

Two towns are at the same elevation and are connected by two roads of the same length. One road is flat, the other road goes up and down some hills. Will an automobile always get the best mileage ...
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How far does a trampoline vertically deform based on the mass of the object?

If a baseball is dropped on a trampoline, the point under the object will move a certain distance downward before starting to travel upward again. If a bowling ball is dropped, it will deform further ...
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Phase Space Flow

Phase space flow shares characteristics with fluid flow such as incompressibility by Liouville's theorem. Extending the similarities one might be curious, does phase space flow have a characteristic ...
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529 views

Three Pendulum Rotary Harmonograph

I'm trying to create a simulation of a three pendulum rotary harmonograph, the one you can see in action in this video or in these instructions. As you can see in the video, there are 2 pendulum with ...
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510 views

Why does motion help you balance on ice skates?

It's almost impossible to balance on a single ice skate if you're standing still. But give yourself just a little forward motion—it doesn't take very much—and it suddenly becomes easy. You can stand ...
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376 views

Physics of scaling up an animal: the neck

Consider an animal like a horse. Now scale its neck longer and longer. How can a giraffe, or even worse a huge dinosaur, raise its neck without the tendons snapping? The dinosaur case in particular ...
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364 views

Will a wave packet undergo dispersion when traveling down a hanging rope?

Suppose I tie one end of a rope to my ceiling and the other end to a spot on my floor directly underneath it. Because the rope has some mass, the tension varies along the rope, from highest at the ...
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208 views

How should I throttle my rocket to reach highest altitude? [closed]

"Real world" problem. Suppose we want to launch a rocket equipped with an engine which can be throttled as we prefer. Suppose also that the amount of fuel burnt per time is directly proportional ...
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2answers
483 views

Find the minimum value of velocity [closed]

Find the minimum value of the initial velocity $u$ of the particle such that the particle crosses the wheel of radius $R$. Details and assumptions $R=2m$ $g=9.8m/s^2$ Neglect air resistance. All ...
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295 views

Liouville's theorem and gravitationally deflected lightpaths

It is customary in gravitational lensing problems, to project both the background source and the deflecting mass (e.g. a background quasar, and a foreground galaxy acting as a lens) in a plane. Then, ...
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692 views

What are the properties of two bodies for their collision to be elastic?

For example, must the shock wave in each body be of a particular form which influences the shape and material properties of the bodies? I suspect part of the the answer is that the objects must be ...
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316 views

The virial theorem and a delta function potential

So the virial theorem tells us that: $2\langle T\rangle = \langle \textbf{r}\cdot\nabla V\rangle$. Now I was wondering what would happen if V has te form: $V(\textbf{r}-\textbf{r}') = ...
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370 views

Mechanical similarity in Landau

I've read this very short paragraph from Landau & Lifshitz's Mechanics (Chap.2, Par.10) (that you can find here) about Mechanical similarity. I was looking for some more detailed explanations of ...
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Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry

Background: Classical Mechanics is based on the Poincare-Cartan two-form $$\omega_2=dx\wedge dp$$ where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, ...
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Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
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9answers
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How to explain independence of momentum and energy conservation in elementary terms?

I'm trying to explain to someone learning elementary physics (16 year old) that linear momentum and energy are conserved independently. I'm not a professional physicist and haven't tried to explain ...
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542 views

The other side of the lever

If I have a lever, but I can see only up to the hinge and not the other half, can I know whether the other half is 1 m long with a weight of 3 kg on it, or 3 m long with a weight of 1 kg on it?
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What sustains the rotation of earth's core (faster than surface)?

I recently read that the earth's core rotates faster than the surface. Well, firstly, it's easier to digest the concept of planetary bodies, stars, galaxies in rotation and/or orbital motion. But, ...
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958 views

Why is tunneling not a classical idea?

There is no tunneling in the case of infinite potential barrier, but there is when we have a finite well. In the classical analog, in the first case we have a particle bouncing between to infinitely ...
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Simple three-body-problem?

Consider the problem of three bodies two of which having mass M, one of them having mass m. Body m is in the middle between the other two, coupled to them by two equal linear springs in rest. Now fix ...
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Is there an inconsistency between Quantum and Classic in probability density of harmonic oscillator ground state?

Consider probability densities for a particle in the lowest energy state of a simple harmonic oscillator. The quantum mechanical probability density peaks near the equilibrium point and extends beyond ...
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Derivation of differential scattering cross-section

I'm trying to follow the derivation of the Boltzmann equation in my Theory of Heat script, but have a little trouble understanding the following: The cross-section $d\sigma$ is defined as: The amount ...
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2answers
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Which is more efficient: a larger wheel or a smaller wheel?

I'm designing a 2-wheeled cart that I plan to rig to a donkey for hauling work around a farm. I'm wondering if there are mechanical advantages to using smaller wheels (like 40 cm diameter) vs. using ...
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Why are infinitesimal rotations commutative, whereas finite rotations are not?

Infinitesimal rotations commute and every finite rotation is the composition of infinitesimal rotations which should logically mean they also commute; but they don't. Why?
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Why $-i\hbar\vec\nabla$ for momentum in quantum mechanics, while $m\vec{v}$ in classical mechanics?

I am a little bit confused when thinking of the momentum representation in QM and CM. In QM, momentum is represented as $-i\hbar\vec\nabla$, while in classical, momentum is represented as $m\vec{v}$. ...
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5answers
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The Z-Torque: how can it be shown intuitively that it does not work?

There is a new kickstarter project that claims to increase torque and power compared to a normal crank on a bicycle (Z-Torque on kickstarter). If this patented (US Patent Number 5899119) approach ...
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3answers
347 views

Poisson structure comes from hamiltonian?

I am interested in studying quantization, but it seems I am lacking the basics of classical mechanics. Any help would be appreciated. I would first like to ask what is necessary to have a ...
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684 views

Why can't we ascribe a (possibly velocity dependent) potential to a dissipative force?

Sorry if this is a silly question but I cant get my head around it.
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Is instantaneous velocity an abstraction?

In introductory analysis, the discussion the derivative emphasizes that while average rates of change are measurable, instantaneous rates of change are a "limiting abstraction". While this makes ...
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The feasibility of a satellite orbiting at a fixed time

I was speaking with some friends of mine, one of whom was an aerospace engineer. He posited the infeasibility of a hypothetical "Margaritaville Satellite" that orbited earth in such a way that ...
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197 views

Galilean, SE(3), Poincare groups - Central Extension

After having learnt that the Galilean (with its central extension) with an unitary operator $$ U = \sum_{i=1}^3\Big(\delta\theta_iL_i + \delta x_iP_i + \delta\lambda_iG_i +dtH\Big) + ...
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Why does higher acceleration minimize a car's fuel consumption?

I generally try to optimize my car's fuel consumption when driving, using my car's real-time MPG gauge and average-trip MPG indicator. Until recently, I believed the slower the acceleration, the ...
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2answers
307 views

The notion called aether

I am trying to learn relativity theory and going through an introductory text on special relativity. I stumbled on the Michelson-Morley experiment. The book claims (accounts) that the result of this ...
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2answers
397 views

How does the distance between two rails effect the speed of a steel ball bearing?

As part of a school science project, I constructed a Rollercoaster using Polyurethane tubing as rails for a steel ball bearing to rest on. In the process of building the coaster I observed that ...
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3answers
529 views

Do we need inertial frames in Lagrangian mechanics?

Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction? My question arose ...
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787 views

Connection between Poisson Brackets and Symplectic Form

Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ {q,p} $ basis are the same as those of the inverse of the symplectic matrix $ \Omega^{-1} $, whereas the matrix elements ...
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Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics?

Books such as Mathematical methods of classical mechanics describe an approach to classical (Newtonian/Galilean) mechanics where Hamiltonian mechanics turn into a theory of symplectic forms on ...
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A pendulum clock problem

Below is a picture of a simple pendulum clock. Suppose that the bob (a rigid disk) on the end of the pendulum can spin without friction about its geometrical axis and is spinning at an angular ...
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2answers
497 views

What are the normal modes of a vertical rope?

Closely related to this question on traveling waves on a hanging rope, I would also like to know what the normal modes are on a rope that hangs vertically, fixed at both ends. Tension in the rope ...
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2answers
180 views

Is it possible to estimate the speed of wind by the sound emitted by a cable of an overhead power line?

I was near ($\approx40m$) an overhead power line and I heard a sound coming from the cables of the power line; I think the sound was made by the vibrations of the power cables due to the wind but I am ...
6
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1answer
437 views

Can a deformable object “swim” in curved space-time? [duplicate]

Possible Duplicate: Swimming in Spacetime - apparent conserved quantity violation It is well known that a deformable object can perform a finite rotation in space by performing deformations ...
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763 views

Particle in a 1-D box and the correspondence principle

Consider the particle in a 1-d box, we know very well the solutions of it. I'd like to see how the correspondence principle will work out in this case, if we consider position probability density ...
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3answers
434 views

Principle of Least Action via Finite-Difference Method

I am reading Gelfand's Calculus of Variations & mathematically everything makes sense to me, it makes perfect sense to me to set up the mathematics of extremization of functionals & show that ...
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510 views

What's the optimal shape for a continuous Galilean Cannon?

A Galilean Cannon is a toy similar to the famous basketball-and-tennis-ball demonstration. You take a tennis ball, balance it on top a basketball, and drop them both. The tennis ball will bounce up to ...