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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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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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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34 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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1answer
61 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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38 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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41 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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116 views

Why is angular velocity of any point about any other point of a rigid body always the same?

I have the following question regarding an ideal rigid body. Firstly, is it always true without any exceptions that the angular velocity of any point about any other point on a rigid body is always ...
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1answer
56 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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60 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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626 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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36 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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In reverse time, do objects at rest fall upwards?

I want to develop a game where time runs backwards, based on the idea that physical laws are reversible in time. However, when I have objects at rest on the earth, having gravity run backwards would ...
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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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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
42 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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Is my boss wrong about our mechanical advantage from our pulley system?

I work on a drilling rig as a roughneck and we had a lecture today (at the office) about mechanical advantage in pulley systems. Now, I know that my boss is well educated in oil drilling, but my ...
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1answer
82 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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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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Why does my door shut faster when the window is open?

I've noticed that if I shut my door when the window is open in a room, the door will tend to shut faster. If I shut the door when the window is closed with a normal force it will not fully close as if ...
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1answer
65 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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2answers
193 views

Why does leaning back make a kicked ball go higher?

It's relatively common knowledge that when one kicks a football, if they lean back the ball usually goes higher, and if they lean forward slightly it goes at a more desirable height. Why does leaning ...
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0answers
8 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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1answer
104 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!
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273 views

Does Newton's first law state something substantive, or is it merely describing a convention?

Newton's first law is often said to define what an inertial frame is - namely, a reference frame in which a body not acted on by a force will move with constant velocity. In other words, a frame where ...
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33 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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Why does dry spaghetti break into three pieces as opposed to only two?

You can try it with your own uncooked spaghetti if you want; it almost always breaks into three when you snap it. I am asking for a good physical theory on why this is along with evidence to back it ...
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101 views

How to reconcile these two approximations?

I'm working on what should be a simple problem (Taylor Classical Mechanics problem 6.23.), but I'm having a tough time reconciling the many ways it can be tackled. An aircraft whose speed is $v_0$ ...
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305 views

Maximal Gravity

I found this interesting problem in Introduction to Classical Mechanics with Problems and Solutions by David Morin: Given a point $P$ in space, and given a piece of malleable material of ...
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1answer
42 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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2answers
142 views

Isolated and non-isolated systems: Momentum?

I'm having a difficult time understanding why two billiard balls colliding is an isolated system, yet a car crashing into a wall is a non-isolated system. Does it really only have to deal with the ...
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4answers
86 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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2answers
129 views

What's the true reason behind thermal expansion?

Thermal expansion is a normal concept everyday. There are 2 explanations: 1, thermal expansion result in stress, then result in deformation 2, thermal expansion result in deformation, then result in ...
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Force as gradient of scalar potential energy

My text book reads If a particle is acted upon by the forces which are conservative; that is, if the forces are derivable from a scalar potential energy function in manner $ F=-\nabla V $. I ...
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Why are position and velocity enough for prediction and acceleration is unnecessary?

In classical mechanics, if you take a snapshot and get the momentary positions and velocities of all particles in a system, you can derive all past and future paths of the particles. It doesn't seem ...
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1answer
59 views

Measuring the pressure in a container without changing it [closed]

Let's say I have a closed, non-transparent metal container with pressurized gas and I'd like to measure the pressure inside without changing that pressure (or changing it as little as possible). The ...
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1answer
399 views

Why is kinetic energy only “often $(1/2)mv^2$”?

I am reading the first few pages of Nakahara and refreshing my memory on physics I learned a while ago as a physics math undergrad. Nakahara defines a field $F$ to be conservative if it's the gradient ...
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Classical mechanics without coordinates book

I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ...
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1answer
138 views

Movement of a cylinder filled with water

Not long ago I was pretty bored at a dinner and I started playing with a water bottle that was not empty: I've been quite interested in its behavior when putted on its side and pushed: the bottle of ...
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2answers
136 views

What indicates if an object will bounce back?

If I throw a small rock (m = 1kg) at a big rock (100kg) the small rock rebounds. Let's say my weight is 80kg, if I would jump into a big rock instead of bouncing back I would move in the same ...
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1answer
58 views

How do you find the tension in the real world? (Given a rope in a pulley system)

I'm well aware of the formula to calculate tension, however, given a real world situation where you have a closed pulley system. How do you measure the force (i.e., tension) required to pull on the ...
3
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2answers
230 views

What's wrong with my calculation of gravitational potential for a uniform sphere?

This is really embarrassing, but I'm not quite sure where I'm going wrong here... Why is this calculation of the gravitational potential inside a sphere with uniform mass distribution incorrect? ...
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2answers
306 views

Any good resources for Lagrangian and Hamiltonian Dynamics?

I'm taking a course on Lagrangian and Hamiltonian Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics. So far at my university ...
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2answers
91 views

Force needed to push a syringe plunger: does one add force associated with downstream back-pressure to frictional plunger force?

I am trying to figure out how much force $F$ is needed to push a syringe plunger. The plunger needs to overcome the friction force $F_1$ and (a much smaller) inertia force $F_2=ma$, giving the total ...
3
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3answers
340 views

In what limit do we *really* get Maxwell-Boltzmann statistics from Bose-Einstein and Fermi-Dirac?

Fermi-Dirac and Bose-Einstein energy occupation number $n(\epsilon)$ in natural units ($[T]=[\epsilon]$) read $$n(\epsilon) = \frac{D(\epsilon)}{e^{(\epsilon-\mu)/T}\pm 1},$$ where $D(\epsilon)$ is ...
32
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2answers
2k views

The Impossibility ( or Possibility) of Solving $N$-Body Problem

One can obtain the solution to a $2$-Body problem analytically. However, I understand that obtaining a general solution to a $N$-body problem is impossible. Is there a proof somewhere that shows this ...
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3answers
128 views

How do we know that the Fourier transform of space is momentum?

How do we know that the Fourier transform of real space $x$ is the momentum $p$ space or for energy and time, receptively? What's the mathematical process and physical logic?
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Are there limits to human/devices perception?

As far as i know, measurement devices present measurements based on something that affects the device's particles, for instance, forces, heat, tension, voltage... My question is, given that every ...
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Describing the motion of a point-mass [closed]

Consider a point-mass moving around a fixed point on a circle with radius $r$ with constant angular velocity $\omega$. At a certain moment of time, the connection is removed, and the point-mass flies ...
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1answer
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Simulating Phase Space Evolution

I am interested in modeling the time evolution of phase-space $\rho(\vec{q},\vec{p},t)$. I have attempted to use Liouville's theorem $\partial_t\rho=-\sum_{i=1}^{3}(\partial_{q_i}\rho)\dot ...
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Marvin the Martian vs. the Death Star: how much energy will they actually need to disintegrate the Earth? [duplicate]

Marvin the Martian vs. the Death Star: how much energy will they actually need to disintegrate the Earth? According to a detailed analysis by Dave Typinski, Marvin the Martian’s Illudium Q-36 ...