Two model energies for a pair of coupled vibrational coordinates

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness

Problem

Two model energies for a pair of coupled vibrational coordinates, in reduced units, are f1(x,y)=x4+y4+2x2+2y2−2xy,f2(x,y)=(x2−1)2+y2.f_1(x,y)=x^4+y^4+2x^2+2y^2-2xy, \qquad f_2(x,y)=(x^2-1)^2+y^2. ((a)) Find the Hessian of f1f_1, show that it is positive definite everywhere, and give the smallest eigenvalue it takes anywhere in the plane. What may you conclude about a minimum a local minimiser finds on f1f_1? ((b)) Find the region of the plane where the Hessian of f2f_2 is not positive semidefinite, and decide whether f2f_2 is convex. ((c)) A minimiser run on f2f_2 returns (1,0)(1,0). Does convexity tell you this is a global minimum? Is it one?

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