Extremal curves in nilpotent Lie groups

TL;DR

This paper classifies extremal curves in free nilpotent Lie groups using Pontryagin's maximum principle, revealing that abnormal extremals are contained in specific algebraic varieties, and extends these results to nonfree cases.

Contribution

It provides an explicit classification of extremal curves in free nilpotent Lie groups and extends the classification to nonfree cases.

Findings

Abnormal extremals are contained in algebraic varieties of a specific type.

Explicit integration of the adjoint equation is achieved.

Results are extended to nonfree nilpotent Lie groups.

Abstract

We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin Maximum Principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the results to the nonfree case.