Treat the potential below as a function on the whole plane

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Calculus of change and accumulation

Conditions of Validity, Ch Differentiability, the implicit function theorem and the licence to change variables

Problem

Treat the potential

U(x,y)

below as a

C^2

function on the whole plane. The strain energy of a shallow buckled strut, with

x

the buckling amplitude and

y

a transverse displacement (both in metres), is modelled by

U(x,y) = \alpha x^4 - \beta x^2 + \gamma y^2 + \delta x^2 y,

with

\alpha = 1.0\ \mathrm{J\,m^{-4}}

\beta = 4.0\ \mathrm{J\,m^{-2}}

\gamma = 1.0\ \mathrm{J\,m^{-2}}

and

\delta = 1.0\ \mathrm{J\,m^{-3}}

. ((a)) Find every stationary point of

U

, classify each by the eigenvalues of the Hessian of

U

, and state which are mechanically stable equilibria of the strut. ((b)) Keeping

\alpha

\beta

and

\gamma

fixed, find the range of

\delta

for which the strut has any stable equilibrium at all, and describe what happens to

U

once

\delta

leaves that range.

Related theory

Lagrangian mechanics

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