On sub‐Riemannian geodesics on the Engel groups: Hamilton's equations

We study the sub‐Riemannian geometry on the Engel group which is a step 3 nilpotent Lie group on . Our main result is to solve the Hamiltonian equations associated with the bi‐characteristic curves and express the solutions in terms of elliptic functions. Our model covers both the Heisenberg group and the Martinet case when setting certain parameters to be zero.