Coordinate System of Coupled Oscillators and 4D Phase Space representation

In summary, the Coordinate System of Coupled Oscillators can be represented in a 4D phase space, where the motion of the system can be visualized by plotting x1+x2 vs v1+v2. The signs in the equations of motion are determined by the kinetic and potential energies of the system.
  • #1
hypernihl
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Coordinate System of Coupled Oscillators and "4D" Phase Space representation

figure.png


So, I've modeled the interaction between two cantilever beams with the kinetic and potential energies shown in the above figure. The cantilevers are very stiff and have a small oscillation amplitude, so they can be modeled as two interacting springs. In the coupling potential epsilon and N > 0.

I'm having a little trouble wrapping my head around the "signs (+ or -)" in the equations of motion derived from the Lagrangian.

Have I set this up right?

All so, I have been displaying the phase-space by potting v1+v2 vs d-(x1+x2).

I'd appreciate any thoughts on this.

Thanks
 
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  • #2
!Yes, you have set this up correctly. The signs in the equations of motion are determined by the kinetic and potential energies of the system, which you have correctly specified in the figure above.As for the 4D phase space representation, you can plot x1+x2 vs v1+v2 to visualize the motion of your system in four dimensions. This will give you a more complete picture of the trajectories of your system.
 

1. What is a Coordinate System of Coupled Oscillators?

The Coordinate System of Coupled Oscillators is a mathematical framework used to describe the behavior of a system of multiple oscillators that are connected or influenced by each other. It allows for the representation and analysis of the collective motion of the oscillators in a coordinated manner.

2. How is the Coordinate System of Coupled Oscillators different from a single oscillator system?

In a single oscillator system, the motion is described by a single set of coordinates, such as position and velocity. In a Coordinate System of Coupled Oscillators, the motion is described by multiple sets of coordinates, each corresponding to a different oscillator in the system. This allows for a more comprehensive understanding of the dynamics of the system as a whole.

3. What is the significance of 4D Phase Space representation in the study of coupled oscillators?

The 4D Phase Space representation allows for the visualization of the motion of coupled oscillators in four dimensions - three spatial dimensions and one time dimension. This representation helps to identify patterns and relationships between the oscillators that may not be apparent in other forms of representation.

4. How is the behavior of coupled oscillators affected by their initial conditions?

The behavior of coupled oscillators is highly dependent on their initial conditions. Small changes in the initial conditions can lead to drastically different trajectories and behaviors of the oscillators. This is known as the sensitive dependence on initial conditions, or the "butterfly effect".

5. Can the Coordinate System of Coupled Oscillators be applied to real-world systems?

Yes, the Coordinate System of Coupled Oscillators has been successfully applied to a wide range of real-world systems, including biological systems, chemical reactions, and mechanical systems. It allows for a better understanding and prediction of the behavior of such systems, and its applications continue to expand in various fields of science and engineering.

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