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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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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79 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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62 views

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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31 views

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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31 views

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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40 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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5answers
394 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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72 views

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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102 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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145 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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17 views

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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51 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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112 views

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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259 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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262 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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173 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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156 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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65 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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79 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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183 views

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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45 views

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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201 views

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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137 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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92 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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106 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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51 views

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 ...
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64 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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64 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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190 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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55 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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38 views

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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83 views

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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65 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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149 views

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 ...
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55 views

Will the bouncing particle exert greater force on the surface?

Imagine elastic collision and no energy is lost from the system. A particle is emitted from the bottom of a box. The box is in inertial motion. The particle hits the top of the box and travels in ...
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1answer
80 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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73 views

Star vibration frequency due to gravitation

I found the following problem on a Classical Mechanics MIT problem set, which is intended to be solved by dimensional analysis: Derive an expression for the vibration frequency of a star of mass M ...
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102 views

Normal Modes of a Suspended Rod with Strings of Fixed Lengths [closed]

A thin uniform rod of length $ 2b $ is suspended by two vertical light strings, both of fixed length $ l $ , fastened to the ceiling. Assuming only small displacements from equilibrium, find the ...
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46 views

If $(q,p)$ to $(Q,P)$ is a canonical transformation, then does this imply $(Q,P)$ to $(q,p)$ is also?

If $(q,p)$ to $(Q,P)$ is a canonical transformation, then does this imply $(Q,P)$ to $(q,p)$ is also, assuming Hamilton's equations hold for the coordinates $(q,p)$? This seems like it should be true ...
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460 views

How do I turn my bicycle?

No doubt a very simple question with an easy answer that's been puzzling me: If I'm riding my bicycle in the $x$ direction with speed $v$ and turn my handlebars I can end up travelling in the $y$ ...
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34 views

Impact strength test of a container filled with liquid

as a university project I have to project(choose material and dimensioning) a liquid container (NaClO, density 1100 g/L), we ...
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1answer
89 views

Work done: kinetic energy or area under F-ds curve?

Starting from $$F=ma = m \frac{dv}{dt} = m \frac{ds}{dt} \frac{dv}{ds} = m v \frac{dv}{ds}, $$ leads to work done = integral of F.ds = integral of mvdv = change in KE. Suppose a variable force is ...
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161 views

Is it possible to write explicitly the exact solution for forced damped harmonic oscillator?

Preamble Consider a damped harmonic oscillator, with his well know differential equation \begin{equation*} m \ddot{x} + c \dot{x} + kx=0 \end{equation*} and let's find the solution that satisfies ...
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46 views

Where does the Lagrangian come from? [duplicate]

It always puzzles me whenever I work on Lagrangian equations. It is easy to see that $L=T-V$ yields the correct equations of motion, but the question is, how do you get to that formula? Is it trial ...
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1answer
72 views

Determining the geometry of the phase space of a system [closed]

How do we check the geometry of the phase space ? I mean in classical mechanics we use position and conjugate momenta as a space of all possible states of the particle. How do we know that this phase ...
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10 views

solve r2 and r3 [closed]

i am working through a balancing of machines question. i cannot figure out how in the text book they get the RHS to equal 16.538 and 10.4045. i think everything needed is included in the photos. if ...
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82 views

What is the initial angular momentum of a rigid body given an offset impulsed force?

What is the imparted angular momentum to a rigid body if the impulse force is offset by a distance $h$ from the center of mass and the imparted momentum from the center of mass is $mv$? For a ...
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1answer
62 views

Reversibility in classical mechanics

I am watching Susskind's 'Theoretical Minimum' videos. At one point in his course on classical mechanics (2nd video if I remember correctly) he affirms that Netwon's second law of motion makes ...
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90 views

Is trajectory the same as an orbit?

Is trajectory the same as an orbit? I wanted to know about gravity assists, but most books I find are talking about different types of orbits and such. Are they related?
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187 views

Maths behind gravity assist

What kind of maths is behind gravity assists and in general the theory of orbits, and how deep does it go? I am just wondering if I know enough prerequisites!