# Two Body Harmonic Oscillator

• maverick280857
In summary, the conversation is about finding a mathematical treatise on the two body simple harmonic oscillator in classical mechanics, without involving Lagrangian or Hamiltonian principles. The request is for resources that focus on energy and momentum rather than quantum mechanics. One suggestion is to use the search term "coupled oscillations -quantum" to narrow down results. The second link provided by Robphy is found to be helpful, but the search for energy discussions is still ongoing.
maverick280857
Hi friends

I would be grateful if someone could point me to a mathematical treatise (on the internet) about the two body simple harmonic oscillator (classical mechanics only, but no Lagrangian/Hamiltonian...just energy, momentum, Newton's Laws).

I am googling right now but all I find is mostly quantum mechanical treatments.

Thanks and cheers
Vivek

Are you talking about something that is essentially two masses on the ends of a spring ?

Yeah kinda.

Try:
http://aix1.uottawa.ca/~jkhoury/oscillations.htm
http://www.scar.utoronto.ca/~pat/fun/NEWT1D/PDF/COUPLED.PDF

(btw, a googling strategy that might help is to include "-quantum" in order to prefer non-"quantum" results.
I googled: coupled oscillations -quantum .)

Last edited by a moderator:
Robphy, thanks very much for the links and the search tip. The second link was quite useful. Do you know anyplace where I can find energy discussions for the system?

## 1. What is a Two Body Harmonic Oscillator?

A Two Body Harmonic Oscillator is a system consisting of two particles connected by a spring and oscillating back and forth around a fixed point, where the motion of each particle is described by a harmonic oscillator.

## 2. What are the equations of motion for a Two Body Harmonic Oscillator?

The equations of motion for a Two Body Harmonic Oscillator are derived from Newton's second law and are given by:

x1''(t) = -ω1^2x1(t) + k(x2(t) - x1(t))

x2''(t) = -ω2^2x2(t) - k(x2(t) - x1(t))

where ω1 and ω2 are the natural frequencies of the individual oscillators, k is the spring constant, and x1 and x2 are the displacements of the particles from their equilibrium positions.

## 3. How is energy conserved in a Two Body Harmonic Oscillator?

Energy is conserved in a Two Body Harmonic Oscillator as the system oscillates, with the sum of the kinetic and potential energies remaining constant. This is because the forces acting on the particles are conservative, meaning they do not dissipate energy.

## 4. What are the normal modes of vibration for a Two Body Harmonic Oscillator?

The normal modes of vibration for a Two Body Harmonic Oscillator are the natural frequencies at which the system will oscillate. They are given by:

ω± = √(ω1^2 + ω2^2 ± √(4k^2 + (ω1^2 - ω2^2)^2)/2)

These frequencies correspond to the two possible patterns of motion for the particles, known as the symmetric and anti-symmetric modes.

## 5. How does the spring constant affect the motion of a Two Body Harmonic Oscillator?

The spring constant, k, affects the motion of a Two Body Harmonic Oscillator by determining the stiffness of the spring and thus the strength of the restoring force. A higher spring constant will result in a higher natural frequency and shorter period of oscillation, while a lower spring constant will result in a lower natural frequency and longer period of oscillation.

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