Potential of coupled oscillators

In summary, To calculate the potential energy of the coupled oscillators in the given picture, we can directly sum the potential energy stored in each mass and spring. This results in the equation V(x_1,x_2) = 1/2 k_1 x_1^2 + 1/2 (x_2-x_1)^2 k_2 + 1/2 k_3 x_2^2. This method is faster than integrating and allows for easier calculation when there are multiple oscillators. To find the force on the left mass, we can imagine each mass's displacement and calculate the potential energy stored in each spring, which can then be summed to find the total potential energy. This thought process allows
  • #1
ehrenfest
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1
[SOLVED] potential of coupled oscillators

Homework Statement


http://cache.eb.com/eb/image?id=2480&rendTypeId=4
How do you calculate the potential energy of the coupled oscillators in the picture with spring constant k_1,k_2,k_3 as the spring constants from left to write?

Homework Equations


The Attempt at a Solution


The force on the left mass is [itex]-k_1 x_1+(x_2-x_1) k_2[/itex] and the force on the right mass is [itex](x_2 -x_1)k_2 +k_3 x_2[/itex] where x_1,2 is the displacement to the right. We integrate w.r.t x_1 and x_2 (and then reverse the sign) to get [itex]1/2 k_1 x_1^2+1/2 (x_2-x_1)^2 k_2+C[/itex] and [itex]+1/2(x_2-x_1)^2k_2 +1/2k_3x_2^2+C'[/itex]. We compare these two find that [itex]C = 1/2k_3x_2^2[/itex] so [itex]V(x_1,x_2)= 1/2 k_1 x_1^2+1/2 (x_2-x_1)^2 k_2+1/2k_3x_2^2[/itex].

My question is how can you do that faster? That is, you should be able to directly calculate the potential without integrating anything. How would you do that? Can you add the "potential of each mass" or "the potential of each spring" or something? That would make it much easier especially when you have many more oscillators!

EDIT: also can someone tell me the thought process that goes into finding the force, say on the left mass? I just found it by kind of guessing and then checking certain cases...
 
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  • #2
anyone?
 
  • #3
ehrenfest said:
anyone?

Imagine that the first mass is diplaced by a distance x_1 to the right (measured from its equilibrium position) and the second mass is displaced x_2 to the right from its equilibrium position.

Then clearly the first spring is stretched by a a value equal to x_1 so its strores a potential energy 1/2k x_1^2.

The second spring is stretched by (x_2-x_1) (if this value is negative, it simply means that the spring is compressed) so it stores an energy 1/2 k(x_2-x_1)^2.

The third spring is compressed by a value equal to x_2 so it stores a potential energy equal to 1/2 k x_2^2

Obviously if you have n masses, the potential energy in each will be

[tex] \frac{1}{2} k x_1^2, \frac{1}{2} k (x_2-x_1)^2, \frac{1}{2} k (x_3-x_2)^2, \ldots \frac{1}{2} k x_n^2 [/tex]
 

What is the potential of coupled oscillators?

The potential of coupled oscillators refers to the ability of two or more oscillators to influence each other's behavior and synchronize their movements. This phenomenon is often observed in physical systems where multiple oscillators are connected through a coupling mechanism.

How do coupled oscillators behave?

Coupled oscillators can exhibit a variety of behaviors depending on the strength and type of coupling between them. They can synchronize their movements, exhibit chaotic behavior, or display complex patterns such as phase locking or frequency locking.

What are some real-world applications of coupled oscillators?

Coupled oscillators have many practical applications, including in mechanical systems such as pendulum clocks and musical instruments, in electrical circuits, and in biological systems such as the synchronization of heartbeats in fireflies.

What factors affect the potential of coupled oscillators?

The potential of coupled oscillators is influenced by several factors, including the strength and type of coupling, the natural frequencies of the oscillators, and any external forces or disturbances acting on the system.

How is the potential of coupled oscillators studied?

The potential of coupled oscillators is often studied using mathematical models and simulations, as well as through experiments in physical systems. Researchers also use techniques such as phase space analysis and bifurcation diagrams to analyze and visualize the behavior of coupled oscillators.

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