Where Can I Find a Mathematical Treatment of the Spherical Oscillator?

[T-7]

Hi,

I wonder if someone can point me in the right direction. I'm after a mathematical [preferably Lagrangian] treatment of the spherical oscillator (an important physical model used, for instance, in quark confinement, or to model atom traps). Like the Kepler problem, I believe, it's a case where a central force leads to a closed trajectory (?).

Could someone suggest some suitable article(s) online, or reliable texts [hopefully not too advanced; I need to be able to follow the mathematics]? I haven't had much success at finding anything of use so far.

Cheers. :-)

 

[AndyW]

Hi there!

The spherical oscillator is indeed an important physical model, and it can be described using the Lagrangian formalism. The spherical oscillator is also known as the isotropic harmonic oscillator, and it is a special case of the more general anisotropic harmonic oscillator.

To start, I recommend taking a look at the Lagrangian for the spherical oscillator, which can be found in many classical mechanics textbooks. The Lagrangian is given by L = (1/2)m(r^2θ̇^2 + r^2sin^2θφ̇^2) - (1/2)kr^2, where m is the mass, r is the distance from the origin, θ is the angle between the position vector and the z-axis, φ is the azimuthal angle, and k is the spring constant.

To find the equations of motion, you can use the Euler-Lagrange equations. The resulting equations are: mrθ̈ = -kθ and mr^2sin^2θφ̈ = 0.

These equations can be solved to find the trajectory of the spherical oscillator. As you mentioned, the central force leads to a closed trajectory. This is because the spherical oscillator is a conservative system, meaning that energy is conserved and the trajectory is closed.

As for resources, I recommend taking a look at the book "Classical Mechanics" by John R. Taylor, which has a good treatment of the spherical oscillator. You can also find many online resources that explain the Lagrangian formalism and its application to the spherical oscillator.

I hope this helps and good luck with your research!

 

[sir-pinski]

Hello,

The spherical oscillator is a fascinating physical model that has many applications in different fields such as quark confinement and atom traps. It is indeed a case where a central force leads to a closed trajectory, similar to the Kepler problem.

There are several reliable resources available online that provide a mathematical treatment of the spherical oscillator. One helpful resource is the article "The Spherical Oscillator: A Simple Model for Atom Traps" by S. K. Adhikari. This article discusses the mathematical formulation of the spherical oscillator using Lagrangian mechanics and also provides a detailed explanation of the physical interpretation of the model. Another useful resource is the book "Classical Mechanics" by Herbert Goldstein, which covers the topic of the spherical oscillator in depth and includes various mathematical treatments.

If you are looking for more introductory material, I would recommend the online lecture notes on classical mechanics by MIT OpenCourseWare. They have a section specifically dedicated to the spherical oscillator, which includes step-by-step derivations and explanations of the equations involved.

I hope these resources will help you in your search for a mathematical treatment of the spherical oscillator. Happy learning!