[coupled harmonic oscillators] old thread- need elaboration

[rsaad]

Hi
Can someone please explain the answer to the following thread? I tried uncoupling the Hamiltonian but to no avail.

https://www.physicsforums.com/showthread.php?t=602106

Thank you.

 

[mfb]

Did you try the proposed rotation?

$x'=\cos(\alpha)x + \sin(\alpha)y$, $y'=-\sin(\alpha)x + \cos(\alpha)y$

Where α has to be determined to uncouple the Hamiltonian.

 

[rsaad]

Yes i tried but I am still getting xy term

 

[mfb]

You could show your work here, this would help to find the error.

 

[pmacias]

Sure, I would be happy to provide some elaboration on the topic of coupled harmonic oscillators. Coupled harmonic oscillators refer to a system in which multiple oscillators are connected or coupled in some way, such that the motion of one oscillator affects the motion of the others. This can occur in various physical systems, such as pendulums connected by a spring or masses connected by a series of springs.

In the thread you mentioned, the discussion is focused on solving the equations of motion for a system of two coupled harmonic oscillators. This can be done by using the Hamiltonian formalism, which involves writing the equations of motion in terms of the system's total energy (the Hamiltonian) and its associated variables (such as position and momentum).

To solve for the equations of motion, one approach is to uncouple the Hamiltonian by finding a new set of variables that decouple the equations and allow for easier solving. This can be done through a coordinate transformation, where the original variables are replaced by new ones that simplify the equations.

However, as mentioned in the thread, this approach may not always work and can be quite challenging. In some cases, it may be more efficient to use other methods such as numerical techniques or approximation methods.

I hope this helps provide some clarification on the topic of coupled harmonic oscillators. Let me know if you have any further questions.