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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474 views

Classical mechanics: Generating function of lagrangian submanifold

I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation. One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is ...
8
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305 views

Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry

Background: Classical Mechanics is based on the Poincare-Cartan two-form $$\omega_2=dx\wedge dp$$ where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, ...
7
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518 views

Mechanical similarity in Landau

I've read this very short paragraph from Landau & Lifshitz's Mechanics (Chap.2, Par.10) (that you can find here) about Mechanical similarity. I was looking for some more detailed explanations of ...
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380 views

Hamiltonian function for classical hard-sphere elastic collision

I'm trying to find the Hamiltonian function for a system consisting of a single particle in one dimension colliding elastically with a wall at x = 0. Everything I've read on the topic (e.g. this ...
5
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64 views

Animating the Bosonic String

I am interested in studying the classical solutions to the Bosonic string in flat 3+1 dim. spacetime by having them rendered a moving picture on a computer. This is partly for fun, and partly to ...
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159 views

What exactly is the relationship between the symplectic 2-form and the frequency of leaves of integrable systems in classical mechanics?

In classical mechanics we equip a differential manifold with a closed symplectic 2-form $\omega$. The symplectic leaves of integrable systems also have a unique frequency, in literature denoted ...
4
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182 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 ...
4
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91 views

References to Mechanics (Classical, Quantum, Statistical) using Time-Scale calculus?

Time-Scale Calculus, is a theory which unifies ordinary (plus fractional and q-) calculus with discrete (and finite differences) calculus. In a sense, in a similar way the Lebesgue integral (or ...
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141 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial ...
4
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66 views

CM: Need to recover the Hamiltonian, knowing conserved quantities and information about the EOM, possibly via action-angle coordinates

Statement of the problem: I have a system with 2 degrees of freedom and I have found two independent conserved quantities, without knowledge of the Hamiltonian. I'm looking for a method to recover a ...
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103 views

Scaling arguments for the Contact mechanics between two elastic spheres

I am studying a bit granular dynamics and I have seen that two spheres of radius $R$ in contact with a contact area of radius $a$ would need an applied force $F$ on this two spheres that is nonlinear ...
3
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29 views

Driven Pendulum

If the point of suspension of a pendulum is driven periodically in the vertical direction , we can derive the equation of motion for the suspended mass to be of the form, $\ddot{\theta}(t) + ...
3
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41 views

What variable is the conjugate momentum for angular momentum?

From the definition of conjugate momentum for a generalized coordinate we get that the conjugate for angular momentum should be proportonal to its integral with respect to time. According to my ...
3
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0answers
72 views

Free energy of coupled classical harmonic oscillators

I'm looking to find the thermodynamic (NVT) free energy of a classical coupled harmonic oscillator system such as the one below: (image taken from ...
3
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0answers
66 views

What are possible explanations for the permeability of balloon rubber, PET plastic and other synthetic materials for carbon dioxide?

Balloons are definitely not gas-tight. Carbon dioxide just leak by the rubber away. A balloon is filled with carbon dioxide. Knot in it. And play. Shrinkage. After an hour or two the carbon dioxide ...
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140 views

Thermalization of coupled classical oscillators

I would like to understand if it is possible to perform an experiment, where a bunch of classical harmonic oscillators (e.g., LC circuits or mechanical pendula) coupled in a simple manner (e.g., one ...
3
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0answers
103 views

Chocolate dynamics

Now I have found a possible model on how to describe chocolate when it is chewed. It has to do with geometrical transformations when a curve $\gamma$ intersects a manifold $M$. The chocolate is ...
3
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0answers
68 views

How coordinate system shifting is related to similarity transformations?

I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?
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46 views

Why and how almost periodic series constitute the algebra of observable of integrable systems?

In the introduction of his book Noncommutative Geometry, p. 42, Connes explains that when a classical dynamical system has enough constants of motions, the motion of the system is almost periodic, ...
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154 views

Obtaining point of application of Ground Reaction Force with use of a hyperstatic load-cell array platform

I'm looking for the theory of an experiment that is giving me a hard time to perform. I have an instrument composed of a rigid horizontal square plate rigidly supported under each corner by a ...
3
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2k views

Forces and torques about the CENTER OF MASS of a physical pendulum

I'm currently stumped by the following situation. Say we've got a rectangular physical pendulum (think ruler with a hole-punch at one end). It's trivial to analyze the motion of the pendulum with the ...
3
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0answers
264 views

Symmetries of separable potential

For separable potential, say $x^4+y^4$, its symmetry are degenerate. Is that a generic case to every separable potential? I will explain my question: The potential $x^4+y^4$ has $A_1, B_1, A_2, B_2, ...
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292 views

Does a thermally expanding torus experience internal stress?

I'm trying to learn continuum mechanics and thermo-mechanics. As we know, heating an object increases the mean atomic distance $a_0$ of the atoms in a rigid body. Let's assume it is a linear elastic ...
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26 views

Time-independence of Hamiltonian of atomic chain

In the first chapter of Atland and Simons book he gives the Hamiltonian of the atomic chain $$ H[\pi,\phi] = \int dx \Bigg(\frac{\pi^2}{2m} + \frac{k_sa^2}{2}(\partial_x\phi)^2\Bigg) $$ After ...
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0answers
34 views

What is the optimal slope for Archimedes screw?

The Wikipedia article has nice image showing how the Archimedes screw work: As I understand, the red balls do not fall down because they are in minima caused by the screw. Because of material ...
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74 views

Is there any physics arguments behind NASA pen joke

I am just wondering are there any reasonable physics arguments behind difficulty making pen for no gravity conditions. My thoughts are that there are many ways to make it working as: Pressurized ...
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17 views

What physical parameters are really being described when a rope manufacturer cites “elongation %”?

I have a precision pulley system sensitive to micron displacements. I am having issues because I need a cord with low tensioning force, but also low stretch. I am getting some weird hysteresis in the ...
2
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58 views

Decoupling of generalized coordinates in lagrangian

Say you have a lagrangian $L$ for a system of 2 degrees of freedom. The action, S is: $S[y,z] = \int_{t_1}^{t_2} L(t,y,y',z,z')\,dt \tag{1}$ If $y$ and $z$ are associated with two parts of the ...
2
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0answers
55 views

How to relate internal energy to atomic motion?

I am trying to conceptualize how atomic motion leads to the thermodynamically-defined internal energy (denoted as $U$ below) through some broad mathematical relationships. I get that the internal ...
2
votes
0answers
48 views

How did Lord Rayleigh find the volume fraction of argon to air?

In order to isolate for pure nitrogen, Lord Rayleigh and his colleagues took some air and removed oxygen, carbon dioxide, and water vapour, leaving behind what he believed to be pure nitrogen. In ...
2
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0answers
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 ...
2
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0answers
56 views

Rheological behavior of chocolate

If someone eats chocolate, the chocolate goes through the following configurations: $\chi_0:$ chocolate is solid and has a smooth Surface everywhere; the Riemann Tensor vanishes on every Point of the ...
2
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0answers
109 views

A question on Lagrangian dynamics an the velocity phase space

I've struggled in the past with understanding why we can treat position and velocity as independent variables in the Lagrangian, but I think I may have finally become a bit more enlightened on the ...
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0answers
50 views

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 ...
2
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0answers
23 views

In supersonic flow, why must area increase in order to have velocity increase?

Based on the area velocity relation: $\frac{dA}{A}=(M^2-1)\frac{du}{u}$ For M > 1, in order to increase u, we must increase A. I'm wondering what the explanation is physically and why it is opposite ...
2
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0answers
62 views

Force and Energy in robots

There are two similar and hypothetical robots that move with wheels powered by motors, Robot A and Robot B. Robot A has a gear ratio of 3:1 (The gear connected to the motor is three times larger than ...
2
votes
0answers
42 views

Interchanging of variation and integration operator for holonomic systems

Meirovitch says in his "Principles and Techniques of Vibrations" (1997) on p.85: In the case of holonomic systems, the variation and integration processes are interchangeable (...) which means ...
2
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0answers
53 views

Hysteresis in liquid–solid-phase transitions such as Agar

I'm wondering how it is possible for a substance to have a significantly different melting point than its freezing point. What physical interaction "locks" a substance such as Agar into the phase that ...
2
votes
0answers
151 views

Hoop rolling inside a circular hole

A hoop of radius $b$ and mass $m$ rolls without slipping within a stationary circular hole of radius $a > b$ and is subject to gravity. Use the generalized coordinates the rotation angle $\phi$ of ...
2
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0answers
93 views

Problem with derivation of phonons in crystal

In this derivation of phonon solutions, everywhere, we are forcefully assuming the wavelike characteristics along the length of the chain. While all we can deduce for finding out the fundamental ...
2
votes
0answers
119 views

Intuition behind the principle of virtual work

To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset ...
2
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0answers
41 views

Gauge formalism in rigid body mechanics

When doing calculations in rigid body mechanics, it is necessary to choose an origin to calculate torques and angular momenta. However, the underlying dynamics does not depend upon the choice of that ...
2
votes
0answers
85 views

Equations of motion for controlled/driven classical systems? Does D'Alembert's principle apply?

I'm puzzled about how to derive the equations of motion for certain classical systems where some entity is controlling some of the DOFs. For example, consider a double-pendulum, with lengths $l_1$ ...
2
votes
0answers
166 views

Story about a mathematician, a dinner party, and the three-body problem

I remember dimly hearing a story, coincidentally also at a dinner party, and I was trying recently to track the details down with no success. I was hoping someone here might have also heard this story ...
2
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0answers
106 views

Interesting Hamiltonian System

The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function. I am wondering if ...
2
votes
0answers
107 views

Difference of the O(N) Non-linear Sigma model and SO(N) Non-linearSigma model

The Hamiltonian \begin{equation} H=J\sum_{i,j}\vec{n}_i\cdot\vec{n}_j \end{equation} is invariant under a global rotation $\vec{n}_i\rightarrow R\vec{n}_i$, where $\vec{n}$ is a $N$ component rotor ...
2
votes
0answers
35 views

Normal modes of two wires fastened together

The problem is to find the normal frequencies of the system formed by two fastened wires of length L, and different mass per unit length. I already wrote the boundary conditions, but I need to know ...
2
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0answers
71 views

Classical/ Quantum mechanical view of magnetic monopoles

Is there any classical/ quantum mechanical proof for the non-existence of magnetic monopoles? Or is it just lack of experimental evidence that has led us to the conclusion that monopoles do not exist, ...
2
votes
0answers
170 views

Trying to solve 2D Toda Lattice Equation with Lax Pair Approach

I am working on this Hamiltonian: $$ H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch ...
2
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0answers
168 views

Collision of Discs and Snooker Kicks

I woke up this morning thinking about spinning discs. Could someone verify whether my reasoning below is correct? Problem 1 Suppose have two identical uniform discs constrained to move in a plane. ...