Sub - Riemannian geodesics on the free Carnot group with the growth vector ( 2 , 3 , 5 , 8 ) ∗

We consider the free nilpotent Lie algebra L with 2 generators, of step 4, and the corresponding connected simply connected Lie group G. We study the left-invariant sub-Riemannian structure on G defined by the generators of L as an orthonormal frame. We compute two vector field models of L by polynomial vector fields in R, and find an infinitesimal symmetry of the sub-Riemannian structure. Further, we compute explicitly the product rule in G, the rightinvariant frame on G, linear on fibers Hamiltonians corresponding to the left-invariant and right-invariant frames on G, Casimir functions and coadjoint orbits on L∗. Via Pontryagin maximum principle, we describe abnormal extremals and derive a Hamiltonian system λ̇ = ~ H(λ), λ ∈ T ∗G, for normal extremals. We compute 10 independent integrals of ~ H, of which only 7 are in involution. After reduction by 4 Casimir functions, the vertical subsystem of ~ H on L∗ shows numerically a chaotic dynamics, which leads to a conjecture on non-integrability of ~ H in the Liouville sense.