Course Outline

Variational Principles

Stationary points for functions on Rn. Necessary and suﬃcient conditions for minima and maxima. Importance of convexity. Variational problems with constraints; method of Lagrange multipliers. The Legendre Transform; need for convexity to ensure invertibility; illustrations from thermodynamics. [4]
The idea of a functional and a functional derivative. First variation for functionals, Euler-Lagrange equations, for both ordinary and partial diﬀerential equations. Use of Lagrange multipliers and multiplier functions. [3]
Fermat’s principle; geodesics; least action principles, Lagrange’s and Hamilton’s equations for particles and ﬁelds. Noether theorems and ﬁrst integrals, including two forms of Noether’s theorem for ordinary diﬀerential equations (energy and momentum, for example). Interpretation in terms of conservation laws. [3]
Second variation for functionals; associated eigenvalue problem. [2]

D.S. Lemons Perfect Form. Princeton Unversity Press 1997
C. Lanczos The Variational Principles of Mechanics. Dover 1986
R. Weinstock Calculus of Variations with applications to physics and engineering. Dover 1974
I.M. Gelfand and S.V. Fomin Calculus of Variations. Dover 2000
W. Yourgrau and S. Mandelstam Variational Principles in Dynamics and Quantum Theory. Dover 2007
S. Hildebrandt and A. Tromba Mathematics and Optimal Form. Scientiﬁc American Library 1985