Local convergence of Newton’s method in the classical calculus of variations

Abstract

Sufficient conditions for a weak local minimizer in the classical calculus of variations can be expressed without reference to conjugate points. The local quadratic convergence of Newton’s method follows from these sufficient conditions. Newton’s method is applied in the minimization form; that is, the step is generated by minimizing the local quadratic approximation. This allows the extension to a globally convergent line search based algorithm (which will be presented in a future paper).

Keywords:

• calculus of variations
• Newton’s method
• local convergence

AMS Subject Classifications:

• 49M15
• 49K15

Acknowledgments

The contributions of the anonymous referees, whose suggestions have resulted in significant improvements in the final version, are gratefully acknowledged.