Questions tagged [anharmonic-oscillators]
Problems concerning oscillators that are not harmonic (i.e. for forces that are not linearly proportional to position).
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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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Solving differential equation in perturbation theory
The differential equation of an anharmonic Oscillator with Newtonian friction is
$$
\ddot{x}+\varepsilon \dot{x}^2+x=0
.$$
The initial conditions of the System are
$$
\begin{align*}
x(0)&=1\\
\...
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Anharmonic effects in crystals, help with intuition
I've been reading a bit about how it is necessary to consider anharmonic effects in crystals if one wants to properly understand things like thermal expansion etc. So for example here:
So the cubic ...
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Perturbation to two-point correlation function for the anharmonic oscillator [closed]
I am trying to answer a question regarding the computation of the first-order correction to the two-point correlation function for the anharmonic oscillator with Lagrangian:
$$ L = \frac{m}{2} \dot x^...
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What is the simplest PDE/ODE/model I can use to understand how nonlinearities can lead to leakage of energy to higher harmonics in an oscillator?
I came across this problem in the study of surface waves in an oscillating cylindrical vessel of liquid.
There are various eigenmodes described using Bessel functions, and energy transfer can happen ...
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Perturbative expansion of energy eigenstates [duplicate]
If we add quartic term in quantum harmonic oscillator,
$$V(x)=\frac{mx^2}{2}+\frac{m^{2}\omega^{3}}{\hbar}\hat{x}^{4}.$$
$$H(\lambda)\,=\,H^{(0)}+\lambda\,\frac{m^{2}\omega^{3}}{\hbar}\dot{x}^{4}\,=\,...
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Nonlinear PDE from Chain of Oscillators
Some years ago, I was reviewing the calculation for the dynamics of limiting case for a chain of springs with transverse oscillations and found a partial differential equation for which I haven't been ...
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Rigorously prove the period of small oscillations by directly integrating
This answer proved that
$$\lim_{E\to E_0}2\int_{x_1}^{x_2}\frac{\mathrm dx}{\sqrt{2\left(E-U\!\left(x\right)\right)}}=\frac{2\pi}{\sqrt{U''\!\left(x_0\right)}},$$
where $E_0:=U\!\left(x_0\right)$ is a ...
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Why is the bending mode of carbon dioxide harmonic?
Here's a simple classical model of a carbon dioxide molecule:
This gif illustrates the "bending mode" vibration.
If the carbon atom moves a small distance $\mathrm{d}x,$ then the springs' ...
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When is this specific Duffing oscillator chaotic?
Assume that I have a Duffing oscillator with no damping and no forcing.
I will be left with
$x''+ω_0^2x=βx^3$
From this website: https://mathworld.wolfram.com/DuffingDifferentialEquation.html , it ...
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Trying to prove chaotic motion from the equation of a nonlinear oscillation [closed]
So I'm given the equation of a nonlinear oscillation:
$x''+ω_0^2x=λx^3$
Assume that $x_1$ and $x_2$ are solutions to the differential equation above.
Therefore;
$x = αx_1+βx_2$
$x' = αx_1'+βx_2'$
$x'' ...
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Elliptic integrals and the classical quartic oscillator
Consider the following Lagrangian in two dimensions:
$L = \frac{1}{2}m\left(\dot{r}^{2}+r^{2}\dot{\theta}^{2}\right)+\frac{1}{2}m\omega^{2}r^{2} + a r^{4}$
I'm interested in studying the radial motion,...
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Does classical simple harmonic motion violate thermodynamics?
If SHM allows for motion to occur forever, we can consider it perpetual motion, does this imply that the second law of thermodynamics is violated? Or does the presence of an external force act on the ...
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Integral partition function of a cubic anharmonic oscillator Energy complex values [closed]
I am interesting in the following integral
$$\int_{-\infty }^{\infty } e^{-\frac{g z^3}{6}-\frac{z^2}{2}} \, dz.$$ Mathematica does not provide any result nor maple either I try to used$$ \text{...
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Why is synchronisation only possible for self-sustaining oscillators
A self sustained oscillator is any oscillator which obeys the following 3 key properties (Balanov 2009):
They do not damp
They are capable of oscillating without being driven by an external force.
...
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Calculate period time from potential energy [duplicate]
The question asks:
The potential energy of a particle in one dimensional space is:
$$
U = \frac{1}{2}Ax^2 + \frac{1}{4}Bx^4
$$
I need to calculate the period time, from the period calculate the ...
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Parametric Resonance Analysis using Perturbative approach
I'm reading Parametric Resonance from Landau's Mechanics Text. A similar calculation is done here. Supposing a parametric oscillator given by
$$\ddot{x}(t)+\omega_0^2(1+h\cos(\gamma t))x(t)=0$$
It's ...
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Analysis of damper having mass
How to do analysis of a non ideal viscous damper (damper having mass)? How to find out an equivalent system for it?(having a ideal damper)
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How to solve SHM questions using Taylor's Series? [closed]
Q) A point particle is acted upon by a restoring force $-kx^3$. The time period of oscillation is $T$, when the amplitude is $A$. The time period for an amplitude $2A$ will be
(A) $T$ (B) $T/2$ (C) $...
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Solution of a polynomial boson problem
I was reading about anharmonicity that can be added to the harmonic oscillator model and I got myself wondering whether it is possible and whether there is a general method to solve (I mean ...
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Why is a phase shift through a time delay not used for damping in vibration dampers?
Why do oscillation dampers use signal conversion through a sufficiently massive electrical circuit (with resistors, capacitors, diodes) to create antiphase, instead of simply shifting the signal in ...
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Decreased period of pendulum
I'm doing an experiment with a physical pendulum and as time passes, the time taken for n cycles is shorter than would be predicted from the period (i.e. the time taken for the first n cycles is less ...
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Poincare section for the duffing Oscillator
I have used the 4th Order Runge-Kutta method in order to estimate the values in which the Duffing Oscillator is chaotic. According to Wikipedia, the Duffing Oscillator is chaotic for values of $\alpha$...
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Asymptotic frequency of nonlinear oscillator $\ddot x = -x-{\dot x}^3$ (speed cubed)
A particle oscillates according to the equation
$\ddot x = -x-{\dot x}^3.$ The positive positions of the particle when it changes direction, $\dot x = 0$, are $x_1,x_2,\ldots$.
I want to show that
$$\...
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Equations of the spherical pendulum in different coordinates [closed]
I am trying to derive the equations of motion of a spherical pendulum, but instead of using the angles of the spherical coordinate system $\theta$ and $\varphi$, I want to use the angles $\alpha$ and $...
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Solve nonlinear, forced and damped Duffing oscillator
I am trying to solve a Duffing type equation by using Van Der Paul's method:
\begin{align}
\ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t)
\end{align}
with $$x(t) = Re[A(t) \...
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Differential equation for the anharmonic oscillator
In my project me and my partner used the engine to constrain the system so we can see the anharmonic oscillations. In our first analysis we get only odd powers in differential equation, so there ...
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Why is a kalimba note anharmonic?
I play a kalimba, and have also recently written a toy program that helps me tune it using a "naively applied FFT" without any sophisticated DSP. The Nyquist frequency is 24 kHz.
The kalimba ...
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Compute probability current from WKB approximation
I struggle to reproduce a calculation from the Appendix of the paper "Anharmonic Oscillator: A Study of Perturbation Theory in Large Order", Physical Review D, 7 (6) 1973, link to abstract.
...
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Anharmonic oscillator without friction
My friend and I are doing a project on physics at the university. We want to describe the non-harmonic oscillator, so we built the system shown in the picture.
The problem is there is friction between ...
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What is the next higher order correction to Hooke's Law for a typical steel spring?
Near equilibrium the potential of a typical steel spring is well approximated by a quadratic function. I'm having trouble finding a reference for the what is the next-to-leading order contribution, ...
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Anharmonic field theory: pedagogical simplifications
Thank you all for so much help as I work through Zee's QFT book. Here I finally have a question about physics instead of math.
Zee makes several comments that the main thing left to do in QFT is find ...
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Landau-Lifshitz skips a step in anharmonic oscillations
In chapter 28 of Landau-Lifshitz Classical Mechanics textbook they try to explain how to get the motion of a particle with the Lagrangian:
$L=\frac{1}{2}m\dot{x}^{2}-\frac{1}{2}m w_{0}^{2}x^{2}-\frac{...
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What are quantum anharmonic oscillators?
I have just started studying about quantum computers (hardware side) and I am really confused about what is a quantum anharmonic oscillator. I have read somewhere that a qubit is the physical ...
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Deriving a model of a point-driven Chladni plate
Please note — this question considers a point-driven Chladni plate, not Chladni's classical experiment. I'm aware of various other questions concerning the latter here on Physics.SE.
As the title ...
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Divergent Energies and Analytical Continuation - Two questions on the inverted harmonic oscillator and the inverted double well
I have two questions on the general topic of energy potentials that diverge at infinity.
First of all, the inverted harmonic oscillator. I found this post on Physics SE, Inverted Harmonic oscillator.
...
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Approximating the energy levels of the anharmonic oscillator using WKB
I got stuck trying to solve this problem:
Given the potential $$V(x) = \frac{m\omega^2x^2}{2}-\beta x^4,\ \beta>0$$ I need to evaluate the deviation of the energy levels from the harmonic ...
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Ladder Operators for a nonlinear oscillator
I was wondering if there was a way to construct the ladder operators for a nonlinear oscillator given by the Hamiltonian $$H=x^2+p^2+\lambda x^4$$ If we were to just calculate scattering amplitudes, ...
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How to obtain large order perturbation series for cubic anharmonic oscillator?
Consider the potential
$$V(x)= \frac{x^2}{2} + gx^3.\tag{1}$$
Then the time-independent Schrödinger equation becomes
$$\left(-\frac{1}{2}\frac{d^2}{dx^2} + \frac{x^2}{2} + gx^3 \right)\psi = E(g) \psi....
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Time period of an anharmonic but periodic motion [closed]
How do I find the time period of anharmonic motion given an expression of force as a function of $x$?
This is the question I was solving:
$$ U(x)=k|x|^3 $$
where $k$ is a positive constant. If the ...
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Why it is not possible to get exact solution for cubic potential perturbation for 1D SHO and we have to use perturbation theory? [duplicate]
Can anyone help me in providing the process of finding exact solution in case of cubic perturbation in 1D SHO, or any suitable resource?
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Why is the ground state energy of a linearly perturbed quantum oscillator always lower than its harmonic counterpart?
I am concerned with a QHO that is linearly perturbed in $x$, i.e.
$$ H = \hbar \omega \left(\hat{n} + \frac{1}{2}\right) + \lambda \underbrace{\left(\hat{b}+ \hat{b}^\dagger \right)}_{\propto \hat{x}}....
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A question about derivation of the potential energy around the stable equilibrium point
I'm learning about harmonic oscillators. In the last lecture my teacher derived the potential energy of a system that has a stable equilibrium point but not a harmonic oscillator.
At the beginning of ...
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Is it possible for a motion to be isochronous (time period is independent of amplitude) but not true s.h.m.? Can an s.h.m. be non-isochronous?
Is it possible for a motion to be isochronous (time period is independent of amplitude) but not true simple harmonic motion? Can a simple harmonic motion be non-isochronous?
Another question I have is ...
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Meaning of "harmonic"
I'm trying to understand the meaning of the term "harmonic". IE, appearing in following sentence of Fluctuation-dissipation relations for stochastic gradient descent
The second relation (...
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How is a quartic oscillator solved in classical mechanics?
Quantum mechanically, a quartic anharmonic oscillator with potential $$V(x)=\frac{1}{2}m\omega^2x^2+\lambda x^4$$ is dealt with perturbation theory- the approximate energies $E_n$ and energy ...
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First-order correction to energy in perturbed harmonic oscillator [closed]
I know, from the perturbation theory, that, if I have the hamiltonian
$$ \hat H = \hat H_0 + \lambda \hat W$$
where $\hat H_0$ is the unperturbed hamiltonian of which I know its eigenvectors and ...
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Time period of an oscillatory motion [closed]
The question:
A particle of mass $m$ is executing oscillation on the $x$-axis. Its potential energy is $U(x)= K|x|^3$, where $K$ is a positive constant. If the amplitude of oscillations is $a$, ...
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Small oscillations in the given potential
The task is to find the period of small oscillations in the potential
$$U=U_0\tan^2{\Big(\frac{x^2}{a^2}\Big)}.$$
I started with finding the stable equilibrium points:
$\frac{dU}{dx}=0$
$2U_{0}\...
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Hooke's full unapproximated law
It is known that the Hooke's law relating the restoring force of a spring to the distance of retraction from the equilibrium position, is only an approximation.
That is, the equation $F=-kx$ is only ...
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