Equilibrium from multivariable potential energy

[carllacan]

Homework Statement

Three masses are disposed in a circular lane and linked with springs (resting lengths Lo). Find the potential energy and, from it, the equilibrium positions. (see image: https://www.dropbox.com/s/evqcspwlj68p5n9/2014-02-05 16.07.07.jpg )

Homework Equations

The Attempt at a Solution

I have no problems finding the potential energy, but I'm not sure how to find the equilibrium positions from it. Should I derive respect each coordinate separately and get three expresions that i have to equate with 0? Or do a triple partial derivative $\frac{\partial V}{\partial \phi_1 \partial \phi_2 \partial \phi_3}$and the equal that to 0?

 

[TSny]

What can you say about the potential energy function at a point of equilibrium?

 

[carllacan]

What can you say about the potential energy function at a point of equilibrium?

That it is a stationary point. However in this case I have three variables, so I'm not sure if it is enough with equating the derivatives to 0 or if I need to use something more advanced like Lagrange multipliers.

 

[TSny]

Stationary point is right. So equating each first derivative to 0 should do it.

 

[paranormal]

Thank you for your question. In order to find the equilibrium positions from the potential energy, you can use the principle of minimum potential energy. This principle states that a system will be in equilibrium when its potential energy is at a minimum. In other words, the system will be in a stable position when the potential energy is at its lowest point.

To find the equilibrium positions in this system, you can use the equations for the potential energy of a spring, which is given by V = 1/2kx^2, where k is the spring constant and x is the displacement from the equilibrium position. In this case, you have three springs, so you will have three equations for potential energy, one for each mass.

Next, you can take the derivative of each potential energy equation with respect to the displacement, set them equal to 0, and solve for the equilibrium positions. This will give you three equations, one for each mass, which you can solve simultaneously to find the equilibrium positions.

Alternatively, you can take the triple partial derivative of the potential energy with respect to each coordinate, as you mentioned in your attempt, and set it equal to 0. This will also give you three equations to solve for the equilibrium positions.

I hope this helps you find the equilibrium positions in this system. Let me know if you have any further questions. Good luck with your homework!