Symmetry and Finite Coupled Oscillators

In summary, the matrix equation of motion for an infinite system of coupled oscillators with identical mass and spring constant k is represented by \ddot{X}=M^{-1}KX. The eigenvectors of the solutions are those of the translation operator, as they commute with M^{-1}K. However, for a large but finite number of coupled oscillators, M^{-1}K may not necessarily commute with the translation operator. To find the eigenvectors of the solutions in this case, it may be necessary to directly diagonalize them.
  • #1
Ibraheem
51
2
For an infinite system of coupled oscillators of identical mass and spring constant k. The matrix equation of motion is [itex] \ddot{X}=M^{-1}KX [/itex]

The eigenvectors of the solutions are those of the translation operator (since the translation operator and [itex] M^{-1}K [/itex] commute). My question is, for the case of a large BUT finite number of coupled oscillators, does [itex] M^{-1}K [/itex] still commute with the translation operator? and if not, is there a way to find the eigenvectors of the solutions, besides directly finding them by diagonalizing?
 
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  • #2
To your first question, "Yes, of course" since you specify "identical" with respect to masses and spring constants, thus there's no positional dependency. Does that also resolve your second question?
 

1. What is symmetry in the context of coupled oscillators?

Symmetry in the context of coupled oscillators refers to the equal and opposite motion of two or more oscillators in a system. This means that when one oscillator moves in a particular direction, the other oscillators in the system will also move in the same direction, creating a symmetrical pattern of motion.

2. How does symmetry affect the behavior of coupled oscillators?

Symmetry can have a significant impact on the behavior of coupled oscillators. When the oscillators are symmetrically coupled, they can synchronize their motion, leading to a phenomenon known as phase locking. This means that the oscillators will all move in unison, creating a stable and predictable pattern of motion.

3. What is the difference between symmetric and asymmetric coupling in oscillators?

Symmetric coupling means that the oscillators are connected in a way that their motion is equal and opposite. Asymmetric coupling, on the other hand, means that the oscillators are connected in a way that their motion is not equal and opposite. This can lead to more complex and unpredictable behavior in the system.

4. How does the number of oscillators in a system affect its symmetry?

The number of oscillators in a system can greatly impact its symmetry. In general, the more oscillators there are, the more complex and varied the patterns of motion can be. However, when the number of oscillators is large, the system can still exhibit symmetrical behavior if the coupling between the oscillators is strong enough.

5. Can symmetry be broken in a system of coupled oscillators?

Yes, symmetry can be broken in a system of coupled oscillators. This can happen due to external forces or disturbances, or if the coupling between the oscillators is not perfectly symmetric. When symmetry is broken, the oscillators may exhibit more chaotic and unpredictable behavior, and phase locking may no longer occur.

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