Questions tagged [anharmonic-oscillators]
Problems concerning oscillators that are not harmonic (i.e. for forces that are not linearly proportional to position).
79
questions
0
votes
0answers
31 views
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 ...
0
votes
0answers
24 views
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 ...
0
votes
1answer
20 views
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)
0
votes
1answer
70 views
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) $...
1
vote
0answers
29 views
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 ...
0
votes
1answer
32 views
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 ...
0
votes
2answers
41 views
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 ...
1
vote
0answers
36 views
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 $\alpha$...
3
votes
4answers
236 views
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
$$\...
0
votes
1answer
66 views
Equations of the spherical pendulum in different coordinates [closed]
I am trying to derive the equations of motion of a spherical pendulum, but instead of using the angles of the spherical coordinate system $\theta$ and $\varphi$, I want to use the angles $\alpha$ and $...
1
vote
0answers
90 views
Solve nonlinear, forced and damped Duffing oscillator
I am trying to solve a Duffing type equation by using Van Der Paul's method:
\begin{align}
\ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t)
\end{align}
with $$x(t) = Re[A(t) \...
0
votes
1answer
72 views
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 ...
4
votes
1answer
383 views
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 ...
4
votes
0answers
73 views
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.
...
2
votes
1answer
74 views
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 ...
1
vote
0answers
59 views
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, ...
0
votes
1answer
48 views
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 ...
1
vote
1answer
109 views
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{...
2
votes
2answers
1k views
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 ...
2
votes
0answers
220 views
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 ...
5
votes
1answer
179 views
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.
...
0
votes
1answer
163 views
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 ...
4
votes
1answer
134 views
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, ...
4
votes
4answers
636 views
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
$$\left(-\frac{1}{2}\frac{d^2}{dx^2} + \frac{x^2}{2} + gx^3 \right)\psi = E(g) \...
0
votes
1answer
128 views
Time period of an anharmonic but periodic motion [closed]
How do I find the time period of anharmonic motion given an expression of force as a function of $x$?
This is the question I was solving:
$$ U(x)=k|x|^3 $$
where $k$ is a positive constant. If the ...
0
votes
0answers
36 views
Why it is not possible to get exact solution for cubic potential perturbation for 1D SHO and we have to use perturbation theory? [duplicate]
Can anyone help me in providing the process of finding exact solution in case of cubic perturbation in 1D SHO, or any suitable resource?
4
votes
3answers
428 views
Why is the ground state energy of a linearly perturbed quantum oscillator always lower than its harmonic counterpart?
I am concerned with a QHO that is linearly perturbed in $x$, i.e.
$$ H = \hbar \omega \left(\hat{n} + \frac{1}{2}\right) + \lambda \underbrace{\left(\hat{b}+ \hat{b}^\dagger \right)}_{\propto \hat{x}}....
0
votes
1answer
73 views
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 ...
1
vote
1answer
934 views
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 ...
2
votes
1answer
491 views
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 ...
2
votes
1answer
82 views
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 $a$, ...
2
votes
1answer
343 views
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}\...
4
votes
3answers
467 views
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 ...
-1
votes
3answers
89 views
Solving ODE equation for classical field [closed]
I would like to solve the following homogeneous, ODE:
$$\left[\frac{d^2}{dt^2} + m^2\right]\phi(t) + \frac{1}{6}\lambda \phi^3(t)=0.$$
I know the solution is
$$\phi(t) = \frac{z(t)}{1-\frac{\lambda}...
0
votes
3answers
1k views
What are the possible limits for when SHM is not applicable for a vertical mass-spring system?
I am not sure if this is more of a maths problem than a physics if so could admin place in the math stack.
So my question is as follows. I have recently been looking at SHM in a spring-mass, as shown ...
-1
votes
2answers
62 views
How can a stiffness of a spring can be $k(x+x^3)$? Compute energy total of the system [closed]
Consider the motion of a particle of mass $m$ attached to a spring of stiffness $k(x+x^3)$ where $x$ is the displacement.
1) How can look such a spring? I thought that the stiffness of a spring ...
0
votes
1answer
137 views
Why should the $\phi^4$ term necessarily cause scattering while a $x^4$ term in anharmonic oscillator only causes correction of energy levels?
Consider an anharmonic oscillator in quantum mechanics, described by the Hamiltonian $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+bx^4.$$ The $bx^4$ term doesn't cause scattering. The effect of this ...
1
vote
0answers
77 views
Asymptotic behavior of canonical perturbation theory for the classic anharmonic oscillator
What do we know about the asymptotic behavior of the perturbative expansion for the classical anharmonic oscillator?
The Hamiltonian is
$$
H = \frac{p^2}{2m}+\frac{1}{2}m\omega_0^2 q^2 +\mu q^4
$$
...
5
votes
2answers
867 views
Pendulum period [duplicate]
In plane pendulum problem, we can calculate its period using elliptic integration.
In SHO problem, we use approximation such that $\theta\ll 1$ and get the period, $2\pi\sqrt{l/g}$.
Is there another ...
0
votes
0answers
5k views
What is the difference between Non-harmonic oscillation, Anharmonic oscillation and Complex harmonic oscillation?
I am just wondering if the words Non-harmonic oscillation, Anharmonic oscillation and Complex harmonic oscillation mean the same thing. If not what exactly is the difference between them?
Since the ...
0
votes
1answer
24 views
Velocity changes due to crystal anharmonicity?
What is the effect of cubic and higher anharmonicities of a phonon hamiltonian on the velocity of phonons?
0
votes
1answer
473 views
Why is pendulum isochronic? [duplicate]
We all know that the pendulum is isochronic, i.e. that it takes the same time regardless of the amplitude is this is less than 20 degrees.
But how do we prove it mathematically?
What happens when the ...
0
votes
1answer
1k views
Why do large angular displacements affect the amplitude of a simple pendulum? [duplicate]
I was conducting an experiment involving the effect of large angular displacements (10°, 20°, ... 90°) on the amplitude of a simple pendulum. So far, I have found that the period of the pendulum ...
2
votes
1answer
1k views
Quantum pure quartic oscillator
It was recently brought to my attention, that there exist analytic solutions for the quantum pure-quartic oscillator with the hamiltonian
$$ \hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{\lambda}{24} \hat{...
-1
votes
1answer
895 views
Simple pendulm motion for larger angular displacement? [duplicate]
What will be the nature of the motion of a simple pendulum for larger angular displacement? Will that be a periodic motion? If so, will the time period increase or decrease?
3
votes
1answer
819 views
A small error in Landau & Lifschitz "Mechanics" (3rd ed.)?
I think I found a small error in Landau & Lifschitz "Mechanics" (3rd ed.). In section 28 (Anharmonic oscillations), they are discussing how to solve the following anharmonic oscillator problem:
...
-1
votes
1answer
225 views
Does increasing the magnitude of the pendulum angle cause its time period to be underestimated? [duplicate]
The pendulum equation states that the time period $T=2π\sqrt{l/g}$. This is based on the small angle approximation where we approximate l
$$\frac{{\rm d}^2 θ}{{\rm d}t^2 }= -\frac{g}{l}\sin θ \...
5
votes
3answers
8k views
Does amplitude affect time period for spring mass system?
I know that with the formula $T=2\pi\sqrt{\frac{m}{k}}$ the time period is not related to the amplitude. However, would amplitude matter if i do this experiment in real life. Would a greater amplitude ...
0
votes
0answers
69 views
How to write classical Hamiltonian $H = \frac{I^{2}}{2}$ in $(p,q)$ variables?
Suppose I have a completely integrable $1$ degree of freedom Hamiltonian $H(I, \varphi) = \frac{I^{2}}{2}$ written in action-angle variables $(I, \varphi) \in \mathbb{R} \times \mathbb{S}$.
What ...
3
votes
0answers
106 views
Papers/books with theoretical data on anharmonic oscillators
I am trying to solve an-harmonic oscillator (with $x^4$ terms) and need theoretical data to compare with my numerical data. I have searched research papers on anharmonic oscillators and few ...