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Questions tagged [coupled-oscillators]

Harmonic oscillators may have several degrees of freedom linked to each other so the behavior of each influences that of the others. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronize. The apparent motions of the compound oscillations typically appears very complicated, but a more economic, computationally simpler and conceptually deeper description follows resolving the motion into [normal-modes].

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Decoupling Linearly Coupled Wave Equations with Potentials

I'm currently working numerically with wave equations and I was wondering if one can always decouple two wave equations, with potentials, which are linearly coupled. The system I'm talking about is ...
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Differences and similarities between phase transitions of Kuramoto model and thermodynamics

I am a math post-graduate (hardly have any modern physics background) and I'm considering the phase transition analysis on complex networks. To my knowledge, the Kuramoto model (see the wiki of ...
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Is there a generalization of mode coupling theory?

I am currently reading a lecture on coupled-modes theory and have a question regarding the ansatz: $$E_\text{tot}=A(z)E_1(x,y,z) + B(z)E_2(x,y,z) \\ H_\text{tot}=A(z)H_1(x,y,z) + B(z)H_2(x,y,z)$$ and $...
Leopold's user avatar
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How to demonstrate in a simple way that this system of differential equations form a damped harmonic oscillator? [closed]

How may I demonstrate in the most simple way that the following system of differential equation form a damped harmonic oscillator ? $$ \dot x = -\alpha_x x - \omega y \\ \dot y = -\alpha_y y + \omega ...
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Does it make sense to talk about individual energies of interacting quantum particles?

Does it make any sense to talk about energy of any one particle in an interacting system? For example if we talk about a system of two coupled quantum harmonic oscillators of same mass and frequency, ...
HypnoticZebra's user avatar
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How to solve the coupled equation of motion? [closed]

there we have the EOM: \begin{align*} \alpha q_{2} + \lambda - \ddot{q}_1=0 \\ \alpha q_{1} + \lambda - \ddot{q}_2=0 \end{align*} and $q_{i}$ is the canonical coordinates. Can I use the Fourier ...
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Coupled oscillators and stability of equilibrium points

My question is about parts (e) and (f). I have found the matrix to equation of motion to be $\frac{d}{dt}\begin{bmatrix} x_1 \\ x_2 \\ p_1 \\ p_2\end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & ...
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Do particles oscillating in coupled oscillation have only normal modes frequency?

If two bodies are coupled and they are performing oscillations, then do they have only two allowed frequencies (normal modes frequencies) with which they can oscillate or do they have a number of ...
Nikhilesh Singh's user avatar
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Why is this called a `Harmonic Oscillator Chain'?

Consider the following general setup: Assume have a chain of atoms (of mass $m=1$) in one dimension interacting with their nearest neighbor through a interaction potential $U$, and which are in an ...
Monty's user avatar
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Normal modes of three masses attached by two springs

I have the following system: I've applied Newton's 2nd Law to the system and I have found the normal modes proceeding as an eigenvalues and eigenvectors problem. I obtained the frequencies $\omega_1^...
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Equations of motion for coupled harmonic oscillators

We just started QFT, and I'm following our professor's notes but there is a passage I do not understand. We are speaking about a system of $N$ coupled harmonic oscillators $y_j(t)$ for $j = 1, ..., N$ ...
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Coupled quantum harmonic oscillator: Decomposition in terms of number basis representation for numerical implementation

Consider the Hamiltonian of a coupled quantum harmonic oscillator \begin{align} \hat{H}&=\frac{1}{2m}(p_1^2+p_2^2)+\frac{m\omega^2}{2}(q_1^2+q_2^2)+\alpha(q_2-q_1)^2 \\&=\frac{1}{2m}(p_1^2+...
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Energy transfer between modes of linear chain of harmonic oscillators?

Consider a one dimensional chain of N classical point masses interacting with harmonic neighbor forces (with periodic boundaries for specificity). If the positions and velocities are prepared in a ...
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Does a chain of classical harmonic oscillators exhibit non-harmonic oscillations?

Consider a one dimensional chain of N classical point masses interacting with neighbor harmonic forces. Is it possible to find initial conditions (positions and velocities) such that non-periodic (...
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Normal modes of a circular array of interacting particles

I want to study the normal modes of an array of $N$ identical atoms placed in a circular lattice. The particles interact among them via Yukawa interaction potential, $$\phi_Y(r)=\frac{A}{r}exp(-r/r_0)....
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Vibrational Modes and Imaginary Frequencies of a Three Spring System

This question is an extension of the one I posted a few days ago: Rigid Body and Two-Spring System and the Lagrangian. I am attempting to find the vibrational modes and their frequencies of an ...
Alex Vaughan's user avatar
2 votes
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How to find particular solution to this system of two masses connected by a spring with constant applied forces? Does an analytic solution exist?

This question comes from review I'm doing on my own, so it's not any homework question. I thought this question would be easy to solve, but I seem to be stuck and I am having second thoughts, so I'm ...
Maximal Ideal's user avatar
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1 answer
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Interpretation of this Hamiltonian

I'm studying a system with the following Hamiltonian $$ H = \frac{1}{2}P^TAP + \frac{1}{2}Q^T B Q$$ where $P,Q$ are canonical variables (4-vectors) and $A,B$ matrices such that $A = A^\dagger$ and $B =...
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Coupled-mode theory and slowly varying envelope approximation

I am facing a situation where I have the following coupled-system equation: $ \dot{U}(z) = i \; M(z) \cdot U(z) \quad ,$ where U is a N-vector and M is a NxN matrix. Now, the diagonal elements of M ...
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If two pendulums connected by a spring both following Simply Harmonic Motion - why do they need the same time dependence?

I was reading a section of Introduction to Mechanics by Kleppner and Kolenkow: where it talks about the same time dependence. I'm not very familiar with this term but was wondering if there was some ...
Emil Sriram's user avatar
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2 answers
73 views

Diagonalizing the tridiagonal matrix for finding the normal mode [closed]

Suppose we have n particles connected by a string tied to fixed ends. In matrix notation the equation of motion can be written as $\begin{pmatrix}2\omega_o^2&-\omega_o^2&0&0&.&.&...
Iti's user avatar
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Coupled pendulum question about equations of motion [closed]

I am working on problem number 2.3 of the Franklin, Powell, Naemi book Feedback Control of Dynamic Systems. The problem uses the simple coupled pendulum system below, where the two pendulum masses are ...
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Diagonalisation of two coupled Quantum Harmonic Oscillators with different frequencies

I was able to diagonalise $H=\hbar\omega a^{\dagger}a+\hbar\omega b^{\dagger} b+\hbar(u a^{\dagger}b+u b^{\dagger} a)$ using the Bogoliubov transformation \begin{equation} a=\cos{\alpha}c_1+\sin{\...
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4 votes
1 answer
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Two springs soldered together, equations of motion

I know there are already several questions about springs in series, but I think this one is different. It is from an exercise from a German book ("Physik mit Bleistift" by Hermann Schulz) ...
Frunobulax's user avatar
1 vote
1 answer
237 views

Why do systems of $n$ coupled oscillators have $n$ normal modes?

Consider a linear system of $n$ differential equations with constant coefficients corresponding to a physical scenario where I have $n$ coupled oscillators (like $n$ masses attached by springs in ...
Mason Giacchetti's user avatar
2 votes
1 answer
123 views

Why is the product of the $L$ and $C$ matrices for coupled transmission lines diagonal?

Background - transmission line $\newcommand{\ket}[1]{\left \lvert #1 \right \rangle}$ A transmission line can be modeled as an infinite sequence of inductors and capacitors: ...
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Derivative term in chain of LC coupled oscillators Hamiltonian

I am taking quantum superconducting circuits course and I cannot recover a formula provided by the lecturer. I want to calculate the Hamiltonian of the following distributed element model of coplanar ...
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What is the relation between Chebyshev polynomials and coupled oscillators?

I have been told that Chebyshev polynomials are key for finding the normal modes of oscillations of a linear chain of coupled oscillators, since they are the eigenmodes of the system. However, I ...
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1 answer
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Nonlinear PDE from Chain of Oscillators

Some years ago, I was reviewing the calculation for the dynamics of limiting case for a chain of springs with transverse oscillations and found a partial differential equation for which I haven't been ...
motherboard's user avatar
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3 answers
114 views

Mass-spring system linear equations: I don't get the last term, shouldn't it be $V=\frac{1}{2}k_3x_{\text{wall}}^2-2k_3x_{\text{wall}}x_2+k_3x_2^2$?

I don't understand the last term in setting up the linear system of equations for multiple mass-spring systems. It is about the last spring in the next example: Source: https://math24.net/mass-spring-...
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Difference between Symmetric and Phase difference?

For a simple coupled oscillator system such as the one here, with equal spring constants and equal masses (with a displacement from equilibrium of $x_1$ and $x_2$), it follows that: $(\ddot{x}_1+\ddot{...
yolo's user avatar
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How to integrate a system with a hamiltonian with non-diagonal kinetic energy

I have the following classical Hamiltonian for two coupled oscillators in the same molecule: $$H=T+V =\left(\frac{p_1^2}{2\mu}+\frac{p_2^2}{2\mu}+k_pp_1p_2\right)+ \left(\frac{1}{2}\mu\omega_1^{2}x_1^...
poisonDartFrog's user avatar
1 vote
1 answer
305 views

Does Hooke's law explain classical wave behavior?

Will Hooke's law $F = -kx$ applied to a large mass-spring grid array such as: provide the full and complete mechanistic explanation for classical wave behavior, including the 2nd order wave ...
James's user avatar
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3 answers
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How to calculate the energy of a spring-mass system considering harmonic oscillation of the normal mode? [closed]

For a spring-mass system, we know that the potential and kinetic energy are $$E_p = \frac{1}{2}ku^2 \text{ and } E_k = \frac{1}{2}m\dot{u}^2.$$ where $k$, $m$ and $u$ are the spring constant, mass and ...
R. Thomes's user avatar
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328 views

Transfer function of system of coupled 2nd order ODE

I am wondering how to calculate transfer function $H(s)$ of system described by 3 coupled differential equations. The pourpose of work is to calculate "Bodedx" diagram ($|H(i\omega)|(\omega)$...
Vid's user avatar
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Why is any arbitrary motion of a coupled oscillator writable as a linear combination of its normal modes? [duplicate]

Consider the following example of a coupled oscillator. Let two identical pendulums, each of length $\ell$ and mass $m$ be connected by a spring of force constant $k$. The system has two normal modes ...
Solidification's user avatar
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1 answer
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How to describe a series of damped harmonic oscilators?

I am looking for textbooks or papers that provide an analysis for a series of damped springs. I am having a tricky time working out the details on my own. I know that if $F=-k\Delta x$ a series of ...
Chair's user avatar
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22 votes
3 answers
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Equations of motion only have a solution for very specific initial conditions

An exercise made me consider the following Lagrangian $$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$ If I didn't make a mistake the equations of motion should be given by: ...
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Coupled Oscillator Period [closed]

I was studying an example of a coupled oscillator the other day, namely two identical masses attached to three springs, the lateral ones of which with the same elastic constant, when I came across the ...
Matteo Menghini's user avatar
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Estimation of number of states to be used to obtain Wigner-Dyson distribution in a chaotic coupled oscilllator

A coupled harmonic oscillator with quadratic coupling - $$ H = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}(x^2 + y^2) + g x^2y^2, $$ is known to be non-integrable, hence chaotic (for reference look at ...
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2 votes
2 answers
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How to perform a Gaussian functional integral?

I'm completely beginner to the quantum field theory and try to learn the basics of functional integrals. However, I could not understand clearly. Could someone please explain the idea with the help of ...
Advaita's user avatar
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1 answer
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Question about the frequency of normal modes in coupled oscillators and their derivation [closed]

Consider a two-mass system that is coupled by three springs, such that: $m_1=m_2=m$; $k_1=k_3=k; k_2=k_{12}$. It can be written in terms of the following coordinates: $\eta_1=x_1-l_1$ and $\eta_2=x_2-(...
agaminon's user avatar
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In a coupled pendulum system, how do pendulums *know* which one is giving its energy, and which one is receiving it?

Picture the system below: If you give the pendulum on the left a push, it will slowly transfer its energy to the one on the right, until it stops completely and the right one is in full swing, then ...
Guilherme Mendonça's user avatar
1 vote
1 answer
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Self-synchronizing and -desynchronizing systems of oscillators

There are biological systems with adaptable frequencies that are able to synchronize their frequencies, mainly individuals (see e.g. reproductive synchrony). In this case, also the phase is typically ...
Hans-Peter Stricker's user avatar
1 vote
1 answer
105 views

Commutation relations for coupled systems

Consider $N$ coupled oscillators whose Hamiltonian is $$\hat{H}=\sum_j^N\frac{\hat{p}_j^2}{2m} +\frac{1}{2}K(\hat{x}_{j+1}-\hat{x}_j)^2,$$ which represents a system like ($K$ is the spring constant): ...
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1 answer
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Normal modes of coupled oscillators

For two pendulums of mass $m_1$ and $m_2$, coupled by a spring of constant k, both suspended by strings of length $l$, the following matrix equality results from their equations of motion: $$ \omega^2 ...
MrStealYourFrog's user avatar
2 votes
1 answer
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Equations of motion of two bodies attached to three springs

I've been tasked with describing the equations of motion of two bodies attached via three springs, as visualized below. Let $x_1(t)$ and $x_2(t)$ denote the $x$-displacements of boxes $m_1$ and $m_2$ ...
user3002473's user avatar
2 votes
1 answer
131 views

A coupled nonlinear dynamical system in four dimensional phase space

I have come across a coupled nonlinear dynamical system given below $$ r\, \ddot{x} + \dot{x} = \sin y~,$$ $$ r\, \ddot{y} + \dot{y} = \sin x~,$$ where $r$ is some real number and $\dot{x}$ denotes $\...
anu's user avatar
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0 answers
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What are normal modes?

In normal modes analysis the differential equations of the system are Fourier transformed and the Fourier monochromatics are found. I think these monochromatics are usually called normal modes of the ...
SimoBartz's user avatar
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-1 votes
1 answer
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Use Hamilton's principle to show expression for $L$ [closed]

I have following diagram I have here to find the kinetic energy and the potential energy. I think that kinetic energy is: $$T=\frac{1}{2} M(\dot{x_1}^2+\dot{x_2}^2)$$ and the potenitial energy must ...
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