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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Detecting absolute motion inside a box

This is not a contradiction and I know it is impossible but still consider a thought experiment by me and point out if something is wrong. See the following picture and then the explanation follows. ...
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535 views

Why does weak equivalence principle say gravity is equivalent to acceleration?

I am told that the weak equivalent principle, that $m_i=m_g$ (inertial and gravitational masses are equivalent) is equivalent to the statement that in a small system you can't tell whether you are in ...
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5answers
719 views

Noether Theorem and Energy conservation in classical mechanics

I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
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2answers
2k views

Potential energy from opposing magnets repelling each other with a gap of 1 mm

I have two powerful rare earth magnets, that are separated by a distance of 1 mm. I applied energy to bring them closer to each other, hence increasing the potential energy. Now, when one of the ...
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4answers
6k views

Blowing your own sail?

How it this possible? Even if the gif is fake, the Mythbusters did it and with a large sail it really moves forward. What is the explanation?
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2answers
527 views

Small oscillations of heavy string

I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points. First of all I wrote down this Lagrangian: ...
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1answer
49 views

Lagrange equation and a force derivable from a generalized potential

I was reading the solution of this exercise and I have a doubt: A point particle moves in space under the influence of a force derivable from a generalized potential of the form $$U(r,v) = ...
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0answers
33 views

Brachistochrone parametric equations

I'm having a bit of a hard time understanding how the parametrized $y$ equation (given below) of the brachistochrone is correct. When these equations are plotted it gives a concave down graph. ...
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1answer
45 views

Angular velocity and instantaneous rotation axis

Let's suppose that we have a cylinder of moment of inertia $I$ rolling on the floor without sliding, moving with linear velocity $v$ and rotating around an axis passing through the center of mass with ...
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1answer
42 views

Amplitude-Frequency curve

Given a resonance curve just like this: Could someone explain to me what the physical meaning of the intersection with the ordinate is? At first glance I would say it has to be $(0 | 0) $ since ...
4
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1answer
115 views

Dimension agreement in canonical transformation

In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical ...
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4answers
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Is there a formula that gives the position of an object depending on the time, but which doesn't allow the object to surpass the speed of light?

I have found these two formulas: $v = at + v_0$ $x = \frac{1}{2}at^2 + v_0t + x_0$ a is the acceleration v is the velocity x is the position t is the time $v_0$ is the initial velocity $x_0$ is ...
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1answer
19 views

Are launch angles relative to observers?

Supposed we have someone on a moving platform which is at constant velocity. Lets say the person launches a mass at some speed relative to the platform an some angle with respect to the platform. Does ...
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1answer
50 views

Show $\frac{\partial T}{\partial \dot q_j} = m_i \dot r_i^T\frac{\dot r_i }{\partial \dot q_j} $ [closed]

This is a basic result in lagrangian mecanics. Let $T$ be the kinetic energy, $r_i$ be the position of the $i^{th}$ particle in the system I need to show $$\frac{\partial T}{\partial \dot q_j} = ...
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1answer
58 views

What is this equation $f^e = f^a - \nabla U$?

Recently in a mechanics class my prof scribbled down something looked like $$f^e = f^a - \nabla U.$$ Where he claimed $f^e$ is the external force on an object, $f^a$ is the applied force on the ...
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1answer
99 views

Can someone explain intuitively how, for a Galilean universe, $A^4$ is equivalent to $\Bbb{R} \times \Bbb{R}^3$?

I am reading Arnold's book on classical mechanics. Obviously, everyone who's studied basic physics feels comfortable with $\Bbb{R} \times \Bbb{R}^3$. This is just a pair $(t,\mathbf{x})$. There are ...
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1answer
46 views

Black hole repulsion mechnism

Schwarzschild radius of black hole is proportional to its mass. From here we can deduce that black hole density getting lower as black hole grows in pace that is inverse to the square of mass. If it ...
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3answers
137 views

Why is it easier to go uphill on a lower gear?

In cars as well as bicycles, when we are on a lower gear, the driving wheel (the one on the wheels) has a bigger radius compared to when on a higher gear. So on a lower gear the bike/car would move ...
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1answer
80 views

Lagrangian vector field expression

The Lagrangian vector field $X_L$ on the tangent bundle is given in page 4 of Marco Mazzucchelli's "critical Point Theory for Lagrangian systems" to be; \begin{equation} ...
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3answers
1k views

Aircraft Level Flight Trajectory

An aircraft climbs to 15000 feet and enters 'level flight' phase. My basic knowledge of physics says that forces on the aircraft at this time are balanced - as seen in this diagram. Would an ...
2
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1answer
27 views

Comparing Brachistochrone curve with a Hypocycloid curve

I want to compare the time that it takes to slide a particle in a frictionless hypocycloid curve, so time would be given by the arclength divided by the velocity So I need first compute the ...
2
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2answers
112 views

Hamiltonian from a Lagrangian with constraints?

Let's say I have the Lagrangian: $$L=T-V.$$ Along with the constraint that $$f\equiv f(\vec q,t)=0.$$ We can then write: $$L'=T-V+\lambda f. $$ What is my Hamiltonian now? Is it $$H'=\dot q_i p_i ...
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2answers
94 views

Do rotation matrices rotate about inertial or body angles? [closed]

I have Yaw, pitch, and roll angles in that order (Euler 321) to apply to a body reference frame in cartesian coordinate system. I want to know what the body reference frame vector coordinates are ...
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1answer
47 views

Particle disintegration (Landau & Lifshitz)

In the particle disintegration problem in the book by Landau and Lifshit(z), it is considered a particle with velocity $\vec{V}$ in the lab frame, which disintegrates into two particles with masses ...
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0answers
56 views

The principle of least action [duplicate]

I have read about the principle of least action. This principle suggests that nature would allow a particle to travel in a path along which the integral of the difference between kinetic energy and ...
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1answer
50 views

Showing time-invariance of Lagrangian with time-displacement operator

I am trying to show that the time-invariance of the Lagrangian of a simple one-particle system implies energy conservation for that system. The first step is, well, to show that the Lagrangian is ...
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1answer
57 views

Boundary of classical and quantum world

So we know that for the really small world we have quantum mechanical behavior and for big things we have classical behavior. But what is the boundary that differentiates the two? If we make a thought ...
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2answers
158 views

Breaking the Laws of Physics? (Walter Lewin rotation experiment)

Lately i have been watching the MIT Physics Lectures from Dr. Walter Lewin. I find his passion while teaching very fascinating and inspiring. Any way, in the end of the lecture about Torque he showed ...
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0answers
46 views

Thermal de Broglie wave length

If we refer to this wiki page thermal de Broglie wave length, we can see there are two expressions. One is derived using equipartition theorem, which makes perfect sense. The other one used $\pi kT$ ...
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1answer
1k views

How did Feynman derive the physics of medallion vs. plate wobble rate?

I am referring to this: Within a week I was in the cafeteria and some guy, fooling around, throws a plate in the air. As the plate went up in the air I saw it wobble, and I noticed the red ...
2
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1answer
213 views

Classical models with unbounded particle number

Is there any classical model which deals with the birth, life and death of particles? What application could it have? I am talking about a 'billiard-ball' kind of model, but the kind in which balls ...
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0answers
50 views

Does the additivity property of Integrals of motion and Lagrangians valid in all situations?

I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of ...
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1answer
32 views

Find out the expression for angular speed in terms of time

Here is the equation that describes the motion of a planet under the gravitational field generated by a fixed star: $$u=\frac el\cos\theta+\frac 1l$$ where $u$ is the reciprocal of the radial ...
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0answers
48 views

Classical Mechanics Help [duplicate]

I'm an undergraduate student majoring in physics. I don't know why but classical mechanics is giving me a lot of problems and I can't seem to grasp the concepts at all. So far we've been doing ...
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6answers
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How could a cord withstand a force greater than its breaking strength?

How could a 100 N object be lowered from a roof using a cord with a breaking strength of 80 N without breaking the cord?? My attempt to answer this question is that we could use a counter weight. But ...
0
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1answer
27 views

There is some attempt to build a magnetic flywheel (reservoir of motion)

We know how flywheel works! There is some attempt to substitute the flywheel-friction mechanism for some magnetic torque ? Exist some mechanism that uses thermal cycle of gases for generate magnetic ...
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0answers
47 views

Motion in a vertical circle and forces in polar coordinates

Suppose a particle with mass $m$ is whirled at instantaneous speed $v$ on the end of a string of length $R$ in a vertical circle. Let $\theta$ be the angle the string makes with the horizontal. I know ...
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1answer
61 views

Simple explanation of first and second class constraints with an example

Can someone give a simple physical example of first class and second class constraints? I mean, if you were giving a classical mechanics lecture for undergraduates, how would you explain this concept ...
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1answer
54 views

Hamilton-Jacobi theory and initial value problem?

Having read through some recent posts regarding the Lagrangian formulation being interpreted into an initial value problem rather than the familiar boundary condition problem we are familiar with, I ...
2
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1answer
76 views

Transfer between translative KE and rotational KE in a rigid body

I have been inspired by some sci-fi cannons that seem to operate by initially spinning up a projectile inside the cannon, and then suddenly firing the projectile out at high speed. Now, I am wondering ...
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0answers
42 views

Terminal conditions and boundary terms in Lagrangian formulations: what do different choices mean?

For the sake of having compact expressions: $$ \left\langle f,g\right\rangle=\int^T_0 f(t)g(t)\,\text{d}t $$ Given some functional: $$ F=\frac{1}{2}m\!\left\langle ...
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0answers
25 views

Unilateral Torque Constraint on the foot-ground interface

I was studying the basics of legged locomotion and came across the unilateral force and torque constraints at the foot-ground interface. I understood the implication of the unilateral constraint on ...
5
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2answers
371 views

Does the $\frac12mv^2$ law apply to quantum mechanics?

Consider the classical Hamiltonian for a spring: \begin{equation} H = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2}kx^2 \end{equation} This is one of those simple cases where when you work out the math we ...
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1answer
47 views

Euler-Lagrange equation with torsion, question on derivatives

Consider a mechanical system, the Lagrangian of which is: $$-L(u,\dot u)=\int\left(\dfrac{\partial^2 u}{\partial x^2}\right)^2\mathrm{d}x$$ This would correspond to a system in torsion, for example. ...
2
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0answers
122 views

Non-conservative Derivation of Lagrangian [closed]

I was previously led to a recent paper by a SE member that did an alternative derivation of the Lagrangian as an initial value problem with two paths rather than the traditional boundary value method. ...
2
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1answer
25 views

Is there a curve for which a particle restricted to move within it under the gravitational force will always exhibit a pure harmonic motion?

A simple pendulum, for example, is not isochronous for large amplitudes (that is, the frequency will depend on the amplitude). So a particle confined in a circumference will not always exhibit a ...
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1answer
53 views

A problem about harmonic oscillators

A ball with mass $m$ and radius $r$ rolls without sliding inside a cylinder with radius $R (R>>r)$, with $\theta <<1$. Find the angular frequency $\omega$ What I Know: There are ...
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2answers
130 views

Symplectic geometry in thermodynamics

There seems to be analogues between Hamiltonian dynamics and thermodynamics given the Legendre transforms between Lagrangian and Hamiltonian functions and all of Maxwell's relations. Poincarè tried to ...
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2answers
209 views

When can phase trajectories cross?

It's said in elementary classical mechanics texts that the phase trajectories of an isolated system can't cross. But clearly they can, for example for the pendulum, the trajectories look like this: ...
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2answers
31 views

Physical meaning of non differentiatiability of $y(t)$ at a point of an elastic medium

Consider two waves $y_1,y_2$ travelling in opposite directions with equations $$y_1(x,t) = A \sin(\omega t - kx) \\ y_2(x,t) = A \sin(\omega t + kx) $$ That create the following standing wave ...