Equilibrium from multivariable potential energy

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Homework Help Overview

The problem involves three masses arranged in a circular configuration connected by springs, with the goal of determining the potential energy and the equilibrium positions based on that energy.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to find equilibrium positions by considering the derivatives of the potential energy function. They question whether to derive with respect to each coordinate separately or to use a more complex method involving triple partial derivatives.
  • Some participants discuss the nature of the potential energy function at equilibrium points, noting that it should be a stationary point and questioning if equating the first derivatives is sufficient given the presence of three variables.
  • Others suggest that equating the first derivatives to zero may be adequate for finding stationary points.

Discussion Status

The discussion is exploring various approaches to finding equilibrium positions, with participants providing insights into the nature of stationary points in the context of multiple variables. There is no explicit consensus on the method to be used, but some guidance has been offered regarding the use of first derivatives.

Contextual Notes

The problem involves three variables, and there is uncertainty about whether additional methods, such as Lagrange multipliers, are necessary for solving the equilibrium condition.

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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 [itex]\frac{\partial V}{\partial \phi_1 \partial \phi_2 \partial \phi_3}[/itex]and the equal that to 0?
 
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What can you say about the potential energy function at a point of equilibrium?
 
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TSny said:
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.
 
Stationary point is right. So equating each first derivative to 0 should do it.
 
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