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.