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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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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67 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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66 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 ...
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1answer
79 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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44 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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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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372 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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55 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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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. ...
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26 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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57 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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141 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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228 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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34 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 ...
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43 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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42 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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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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56 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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89 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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2answers
48 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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2answers
107 views

Idea of integrable systems

I do not quite understand the idea an integrable dynamical system. Does it mean that the EOMs are analytically and exactly solvable? What are the necessary and sufficient conditions such that a system ...
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173 views

How to formulate variational principles (Lagrangian/Hamiltonian) for nonlinear, dissipative or initial value problems?

Although this questions is very much math related, I posted it in Physics since it is related to variational (Lagrangian/Hamiltonian) principles for dynamical systems. If I should migrate this ...
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198 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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46 views

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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Torque vs Moment

I was wondering, why in Newtonian physics torque is called "torque" while in static mechanics they call it "moment"? I prefer by far the term "torque", for not only it sounds strong, but also ...
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53 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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127 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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186 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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35 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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2answers
1k views

Integrals of Motion

Landau & Lifshitz write on the first page of chapter 2 of their Mechanics book (p.13) The number of independent integrals of motion for a closed mechanical system with $s$ degrees of freedom ...
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34 views

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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1answer
49 views

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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25 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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57 views

Understanding incompressibility (of rubber or viscoelastic material)

Literature gives a lot of explanation why rubber is incompressible. However, I still need some thinking to understand physical behavior of rubber or any such material. Often, incompressibility is ...
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37 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?
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61 views

Potential Energy of Interaction Between a Sphere and a Particle Formula Derivation [closed]

A sphere of radius R has density described by ρ=ρ(r). Derive equation for pontetial energy of interaction between the sphere and some point particle of mass m which is at distance r from the center of ...
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1answer
145 views

Langevin equations in translational and rotational direction

I want to describe the following system. A bead is connected with a tether. There is a force $\vec{F}_{up}=F_{up}\hat{y}$ that acts on the bead. The tether acts with a force on the bead, this force ...
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1answer
43 views

Computing the gravitational force on a planet in a particular system [closed]

I have a system of four planets moving in a 2D plane. I'm trying to write some code (C++) to find the positions of these planets at time t=3. I'm probably going to attempt this via a leapfrog ...
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2answers
83 views

How to analyse this mass-spring system

I'm trying to analyze this mass-spring system -- i.e. write down the differential equation governing it. As you can see, there is a block of mass $m_1$ attached to a wall by an ideal spring of ...
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1answer
180 views

Can a thrown egg chip (or break) a car windshield?

Is it possible to throw an egg with such speed that a car windshield will chip (just like with stone chips?) I have searched around for existing research in the area and have found that the impact ...
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1answer
61 views

Given potentials, how does one find conserved quantities using Noether's theorem?

I've been asked to find the conserved quantities of the following 3D potentials: $U(\vec{r}) = U(x^2)$, $U(\vec{r}) = U(x^2 + y^2)$ and $U(\vec{r}) = U(x^2 + y^2 + z^2)$. For the first one, ...
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1answer
41 views

What will happen to the center of mass of the human body when a person carries a weight with one hand?

What will happen to the center of mass of the human body when a person carries a weight with one hand a briefcase for example. Wouldn't the center of mass move horizontally towards the side carrying ...
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23 views

Would the torque required by a motor differ depending on where it connects to a frame?

Given a motor attached to a flat surface, aligned to the axis of a laptop screen and connected to the screen via an L shaped arm which is also connected to the motor shaft Would the torque required ...
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68 views

Stability of Mathieu's equation and parameteric resonance

I am given the following equation (Mathieu's equation) in my subject of Numerical Analysis : $$ \frac{d^2 x}{dt^2}=-\omega^2(1+\epsilon\cos(t))x $$ I am supposed to find those frequencies $\omega$ ...
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1answer
37 views

Elementary proof of the minimum number or parameters needed to uniquely identify a force-torque (aka wrench) in 2D vs. 3D

Since the term force-torque (aka wrench vector) is probably more common in Robotics than in Physics, let's try to start with a definition of what is sought: a force-torque is a parsimonious set (well, ...
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74 views

Symplectic structure and isomorphisms

In his book Mathematical Methods of Classical Mechanics, V.I. Arnold writes To each vector $\xi$, tangent to a symplectic manifold $(M^{2n},\omega^2)$ at the point $\mathbf{x}$, we associate a ...
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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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36 views

Motion Integrals of a Particle in a Force Field

I am trying to wrap my head around the following problem: A point particle is moving in a field, where its potential energy is U=-α/r. Find first motion integrals. In our university we have no ...
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1answer
38 views

Does this massless spring affect the system?

I have to write out the differential equation modelling this system: There's a mass connected to a wall with a spring of spring constant $k_1$, sitting on a frictionless surface, with another spring ...