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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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 ...
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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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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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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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54 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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62 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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106 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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58 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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86 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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1answer
35 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 ...
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56 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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64 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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2answers
128 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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65 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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2answers
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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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69 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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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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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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34 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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54 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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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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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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30 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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47 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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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 ...
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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 ...
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60 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. ...
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375 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
27 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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66 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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128 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. ...
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156 views

Is the strength of a muscle proportional to its cross-sectional area?

I have a question that is partially related to at least a couple of old questions: this one and this other. My question is specifically focused on the following point: why should the strength of a ...
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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 ...
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84 views

Explanation of force amplification inside a solenoid

For a system being actuated by a motor, the force can be amplified by gearing. The energy is being used for force instead of distance, so it produces more torque but moves slower. For a system being ...
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3answers
52 views

Where is the energy stored in destructively-interfering waves?

Let's say we have two waves moving along a string. One of them is represented by the function: $$f_1(t)=\sin(\omega t)$$ The other one is represented by a function: $$f_2(t)=-\sin(\omega (\tau-t))$$ ...
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3answers
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origin of the major symmetry property of the elasticity tensor

In linear elasticity theory the stress tensor $\sigma$ is related to the strain tensor $\epsilon$ via the elastic tensor $C$. Specifically $$ \sigma_{ij} = C_{ijkl} \epsilon_{kl} $$ Because $\sigma$ ...
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73 views

Independence of position and velocity in Lagrangian from the point of view of physics?

I would like to continue discussion from my previous post on time dependence of lagrangian Time dependence of the Lagrangian of a free particle?. I have also read this old post Why does calculus of ...
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2answers
64 views

Pendulum point in polar coordinates for Lagrangian

So I'm really stumped with this. I have a particle in a cone, like pictured. The particle orbits the z axis on the dotted line for $r$. So knowing that $\alpha$ and $r$ remain constant in this ...
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3answers
97 views

Time dependence of the Lagrangian of a free particle?

I am working through Landau's book on Classical Mechanics. I understand the logic and physics of isotropy and homogeneity of space-time behind the derivation of the Lagrangian for a free particle, but ...
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4answers
232 views

Help understanding what the Hamiltonian signifies for the action compared with the Euler-Lagrange equations for the Lagrangian?

Consider the Lagrangian for a simple harmonic oscillator \begin{equation} L (x,\dot{x}) = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 \end{equation} Obviously we have \begin{align} \frac{\partial ...
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Is there a typology of different fundamental physical objects?

As I understand it (and of course, I may be wrong!).... In classical mechanics, all objects are basically the same in the sense that They are composed of atoms bunched together. These atoms occupy ...
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54 views

Given an initial push, is work done on an object infinite in a hypothetical empty universe?

Consider a hypothetical empty universe containing a single object. Given an initial push, will the work done by the forever moving object be infinite?
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212 views

Definition of generalised coordinates?

I think the definition of generalised coordinates is something along the following lines: A set of parameters that discribe the configuration of a system with respect to some refrence ...
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1answer
191 views

Why does deflating baloon spurting through the air make circular motion? [duplicate]

When you inflate a balloon and then let it go again, it will fly through the air in an unpredictable motion. My kids (1 and 3 year old) love watching this. At some point my oldest asked how it worked ...
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39 views

Calculating/estimating heat transfer losses for hot air balloon (lantern)

I'm trying to build a flying lantern / hot air balloon that flies as close to hovering as possible (as opposed to up-up and awaaay). To see if this is feasible I'm trying to simulate as much as I can ...
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What justification is necessary for convolutional variational principles to be considered legitimate?

I recently asked a related question and was interested in why/or why we cannot use convolutional variational principles in practice or in theory. Summarizing the points I made in the earlier post: ...
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Can individual forces be regarded as momentum flows? [closed]

Net force on an object can be defined in two ways equivalently (from a classical point of view): $$\vec{F} = m\frac{d\vec{v}}{dt}=\frac{d\vec{p}}{dt}$$ Looking at the last expression (definition in ...
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28 views

A central force which enables a torque on a sphere - is it still conservative?

Consider the following example: Two spheres (one big, other small) standing vertically on ground. At first, the small sphere is on top of the big sphere. Then, it starts to roll w/o slipping to ...
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45 views

Velocities of points along an inextensible string

It is a well known constraint that velocities of points along an inextensible taut string or rod is constant. This is, for instance of use in the following problem: If a rod slides along the wall ...
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How does the earth do a negative work on a static body? [closed]

If a body is in rest and the earth acts with a force on it 10 N Is there a negative work done by the earth though the body doesn't move? how? does it have a common thing with potential energy?