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  • Print publication year: 2019
  • Online publication date: October 2019

18 - Integrability of the Sub-Riemannian Geodesic Flow on 3D Lie Groups


In this chapter we show how to find certain first integrals, for Hamiltonian systems on Lie groups, that are automatically in involution with each other and with the Hamiltonian. This theory will be used to prove that the Hamiltonian system for normal Pontryagin extremals for rank-2 left-invariant sub-Riemannian structures on three-dimensional Lie groups is completely integrable.