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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Locally accessible dimensions of configuration space

I am reading a book called "Structure and Interpretation of Classical Mechanics" by MIT Press.While discussing configuration space and degrees of freedom,the authors remark the following: Strictly ...
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Magnetic Field and Flow of Vector Potential

I am sorry, when my question is not really concrete, but here we go. Consider the Hamiltonian function $$H(x, \xi) = \frac{1}{2m}\bigl|\xi - eA(x)\bigr|^2$$ corresponding to a charged particle in a ...
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Lagrangian mechanics and initial conditions vs boundary conditions

It bothers me that many basic books on the classical mechanics don't discuss the following difference between "Newton's laws" and the "Principle of stationary action". Newton's laws can predict the ...
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On the isotropy of materials

Good morning. I am working on Honeycomb structures and first of all I would like to understand whether it is Isotropic or not, and , if the latter holds which kind of anisotropy it has. How to do it? ...
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Classical Limit of the Quantum Harmonic Oscillator

The classical harmonic oscillator obeys an arcsine law in that the distribution of positions of the particle over a single time cycle is proportional to $\frac{1}{\sqrt{A^2-x^2}}$, $A$ being the ...
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50 views

Is Liouville's theorem valid for dimensionally restricted systems?

Liouville's theorem states that the phase space volume of a system is conserved over time. Intuitively, this seems to imply that if a system is at some time constrained to, say, a curve in phase ...
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64 views

Does sound have a “louder” direction?

I have a question about the propagation of sound waves. We have two TV's in our house that are almost right on top of each other. One is located on the first floor and the other one is located on ...
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34 views

Under what circumstances will a particle in a paraboloid oscillate harmonically around the lowest point? [on hold]

A particle of mass $m$ is moving in the gravitational field $g$. It is confined to move frictionlessly on the surface of the paraboloid $z(r,\phi):=a\cdot r^2$. Here, $a>0$ is a constant. The ...
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Calculating the flex of a solid bar under force

I want to calculate how much a solid bar will flex when force is applied to it. The set up looks like this: The rod (in green) rests on two stationary points, and the force is applied in the center ...
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35 views

Can center of mass move without any force?

For instance, consider a weight on one end of the ring. Assume that the ring has negligible mass compared to the weight. When the weight splits into two, moves around the ring and recombines at the ...
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Why does the Stern–Gerlach quantum spin experiment conflict with classical mechanics?

My understanding of the Stern–Gerlach experiment is that neutral (0 total charge) particles are sent through a non-homogeneous magnetic field, with the expectation that the field will push that ...
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40 views

How much time it takes for a car to pass a truck [on hold]

The driver of a car wishes to pass a truck that is traveling at a constant speed of $19.7~m/s$ . Initially, the car is also traveling at a speed $19.7~m/s$ and its front bumper is a distance $24.1~m$ ...
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How do I calculate the speed of the air particles flowing out of a balloon? [duplicate]

I am trying to find out what kind of force would a leakage in a balloon cause. What i used is F = (mass flow)speed = (air density)(surface of leakage hole)*speed. I don't know how I could calculate ...
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52 views

Springs at an angle [closed]

I'm trying to find the equation of motion for the following system: This is how I proceeded: Let's call the length of the hypotenuse $s$. Then, $$F = 2 \sin{\theta}\cdot-k(s - l_o) = -2kx ...
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23 views

Acceleration in a system equal for every body — why?

A 1 kg cart can slide frictionlessly on the table. The black weights each weigh 1 kg. The pulleys are frictionless. The task is to determine the acceleration of the cart. For the left-most ...
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25 views

Fluids in a U-shaped Tube

One of the users asked a question about the Fluids in U-shaped Tube. I was wondering and I tried to imagine that the membrane is fixed and the left side is filled up until $h_1=h_2$. So my question is ...
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3answers
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What relative masses are required for them to collide n times in this scenario?

Consider two masses, m and M, where M>m. They begin at rest on an infinite frictionless surface that is flat in one direction and sloped in the other direction. Mass m is placed a little bit up the ...
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What is the equation of motion for a driven spring?

A spring of length $l$ and spring constant $k$ is suspended vertically with an object with mass $m$ attached at the bottom. If you take the top of the spring and oscillate it such that its ...
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Parametric impulse on driven, damped oscillator

I've been thinking about driven harmonic oscillators recently. I know how to calculate their response to a sinusoidal drive, and their response to an impulse or more generally an arbitrary drive via ...
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Difference between Kinematic, Dynamic and Differential constraints in robot path planning [closed]

I have started to work on planning motions for aerial robots. I would like to know the simple difference between kinematic, dynamic and differential constraints, for ideas presented in different ...
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31 views

Born-like measuring rule in classical experiments

this 2011 paper "Born's rule from measurements of classical signals by threshold detectors which are properly calibrated" by Khrennikov investigates the theoretical possibility of Born-like ...
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3answers
166 views

Is the polar coordinate system non-inertial or inertial?

Consider a car driving around in a circle lying in the plane and suppose we were interested in determining its acceleration as measured by an observer stationary on the "ground" or whatever. ...
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28 views

Computing the angular momentum in spherical coordinates

How to compute the angular momentum of a particle in spherical coordinates? It's given by: $$x_1=r\cdot\cos(\phi)\cdot\sin(\theta)$$ $$x_2=r\cdot\sin(\phi)\cdot\sin(\theta)$$ ...
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Examples of (classical) measurements that are not independent?

What are some simple examples of measurements that are not statistically independent, i.e. with nonzero covariance? I'm looking for real examples that might reasonably come up in an undergraduate ...
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75 views

How is the special theory of relativity observed for these types of clocks?

I'm trying to understand Special Theory of Relativity through reading Feynman's lectures. In chapter 15 Feynman gives example of clock: rod of 1m length with mirrors at the ends. Light goes from ...
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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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When will be a suspended (not at the center of the mass) symmetric rotating gyroscope in stable or instable position?

The original question is what is in the title. I'm not sure about the answer so here is my solution, please correct me If I am wrong: It is known for a gyroscope what is in a homogenous gravity field ...
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33 views

Conservation of probability in phase space flow

In J.Binney's notes on classical mechanics, under the section 'Liouville's theorem', he states that (paraphrasing): the conservation of probability requires that $\frac{df}{dt} = 0.$ where $f$ ...
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Why does this ball have potential energy at its lowest point?

A ball of radius $r_0$ starts from rest at point $A$ on the inside of a track of radius $R_0$. The question is what will its speed be when it reaches the lowest point of the track, point $B$, ...
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237 views

Is it possible to determine the outcome of any impact knowing only the ratio of masses? [duplicate]

In elastic collisions in 2-D if two balls $A$, $B$ ($m_A = m_B$, $R = 1$) have equal mass we can determine in advance the outcome of the collision. If cue-ball $A$ impacts object-ball $B$ (at rest) ...
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46 views

Rolling without slipping and friction

When a ball rolling without slipping along an inclined plane reaches the bottom, it has a linear velocity $v$ and angular velocity $\omega\ =v/r$ at the bottom. Then it continues its motion on the ...
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158 views

Integrability of the many body problem

In classical canonical perturbation theory of many degrees of freedom we encounter the problem of small divisors when attempting to find a solution for the generating function of the canonical ...
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147 views

Shouldn't motion be represented as a Taylor series rather than a finite sum of functions or a polynomial? [closed]

Since the change in velocity of an object at rest prior to time $t_{0}$ implies a change in acceleration — that is, let's postulate, $ \mathbb{P} $, the object would have remained still, so there was ...
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47 views

What are the necessary/sufficient conditions for a system to be Hamiltonian/non-Hamiltonian?

I searched for a definition of Hamiltonian system on Huang and Tuckerman text but have not found anything precise. So intuitively I suppose: Hamiltonian system= a system which admits a complete ...
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35 views

Momentum and Kinetic Energy Conservation in Inelastic Collision

Let a ball fall freely to the ground, hit and bounce back. Assume mass of the ball does not change during this. Since momentum is conserved in all collisions, $p_i = p_f$ In this example, the system ...
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Prerequisites for classical mechanics by Susskind

So I am an undergraduate in Electrical Engineering. We had a course on Physics in our freshman year which is equivalent to Classical Mechanics I as taught in MIT. I am interested in studying advanced ...
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Density of states and anisotropic distribution functions

We consider a $3D$ dynamical system. Its distribution function is given by the function ${ (\mathbf{x},\mathbf{v}) \mapsto f (\mathbf{x},\mathbf{v})}$, so that $$ \mathrm{d}^{3} \mathbf{x} \, ...
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What determines the bounce time of an elastic ball?

Consider an elastic ball is bounced off a hard flat surface. I would like to reconcile two different answers to the question "how does the contact time between the ball and surface depend on the speed ...
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1answer
23 views

Find angular velocity of motor

I'm quite bad at this, but I'm trying to change that and I need some assistance. Please bare with me while I attempt to explain what I'm trying to figure out and correct me where I'm wrong. Basically ...
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68 views

Elementary questions about where energy is stored in solids

I have the following question, I have no relation to the study of Physics in any way, but the question has been teasing me for some time. Please accept my poor physical terminology. Here we go... ...
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71 views

Is temperature affected by gravitational potential?

Ok, I feel a bit silly asking this. I'm asking in relation to this question here on the molecular basis of hydrostatic pressure in a gas. There's been quite a bit of discussion and one of the ...
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Canonical transformation from Hamiltonian without external source to Hamiltonian with external source

Let a system with time-independent Hamiltonian, $H_0(q,p)$ be subjected to an external oscillating field $E_0\sin(wt)$, so that the Hamiltonian becomes $H=H_0(q,p)-qE_0\sin(wt)$. Find a canonical ...
2
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55 views

Lagrangian under time transformation

Given a Lagrangian $$L(q,\dot{q},t)=\sum_{ij}a_{ij}(q)\dot{q}_i\dot{q}_j-V(q_1,q_2,\cdots,q_f)$$show that under a time transformation $t=\lambda T$ ($\lambda$ = constant), the invariance of ...
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2answers
44 views

Conservation of energy when the Lagrangian includes a potential function

When proving that the homogeneity of time leads to the conservation of energy, (This is the proof from Landau for the case when there is no field present.) (Uses the Einstein's summation ...
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2answers
146 views

Trouble with Landau & Lifshitz

Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
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1answer
42 views

spherical phase space dynamics

I have a hamiltonian of the form $$H(\phi,z) = (1-z^2)\cos(2\phi) + \chi z^2$$ with position $\phi$ and conjugate momentum $z$. It has this form provided that $z \in [-1,1]$ and we have natural ...
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How do I prove that frequencies that are irrationally related lead to quasi-periodic motion?

Consider the equation: \begin{equation} \dot{x} = Mx, \end{equation} where \begin{equation} M = \begin{pmatrix} i\omega_1 & 0 & \cdots & 0 \\ 0 & i\omega_2 & \cdots & 0 \\ ...
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Time reversal in simple *solution* to equation of motion

Consider the solution to the equation of motion for a particle with a constant acceleration: $$ x(t) = x_0 + v_0t + \frac{1}{2}at^2.$$ If I let $t \rightarrow -t$, then the equation becomes: $$ x(-t) ...
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35 views

Direction of velocity confusion on inclined plane

In Taylor's book Classical Mechanics, pg. 259, he works through the following example: Consider the following block and wedge system: The block ($m$) is free to slide on the wedge and the wedge (mass ...
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How does temperature in a solid sphere change with time when moving through a gas?

I'm interested in the following problem: There is a solid sphere with radius $r$ and mass $m$ at temperature $T_{s0}$. It is moving at velocity $v_s$ through a gas of temperature $T_g$. How does the ...