# Pontryagin maximum principle, (co)adjoint representation, and normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups

## TL;DR

This paper develops a foundational approach combining Lie group theory, (co)adjoint representations, and the Pontryagin maximum principle to identify normal geodesics and optimal controls for left-invariant (sub-)Finsler metrics on Lie groups.

## Contribution

It provides an independent theoretical foundation for using geodesic vector fields to find normal geodesics and optimal controls in (sub-)Finsler and (sub-)Riemannian geometries on Lie groups.

## Key findings

- Derived methods for geodesic vector fields on Lie groups.
- Established connections between Pontryagin maximum principle and geodesic equations.
- Applied the framework to specific (sub-)Riemannian cases.

## Abstract

On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally optimal controls in (sub-)Riemannian case, as well as some their applications.