We study the sub-Riemannian structure determined by a left-invariant distribution of rank n on a step-2 simply-connected nilpotent Lie group G of dimension n(n + 1)/2. We describe a transitive group action that leaves invariant the sub-Riemannian structure. By using geometric optimal control theory techniques, we derive necessary conditions for the length-minimality of the sub-Riemannian geodesics. We perform an integration process for the associated Hamiltonian system that yields explicit expressions for the extremal curves and the corresponding sub-Riemannian geodesics, the obtained formulas allow the complete parametrization of the exponential mapping in terms of algebraic invariants of the problem.
A Hamiltonian approach is presented to study the two dimensional motion of
damped electric charges in time dependent electromagnetic fields. The classical
and the corresponding quantum mechanical problems are solved for particular
cases using canonical transformations applied to Hamiltonians for a particle
with variable mass. The Green's function is constructed and, from it, the
motion of a Gaussian wave packet is studied in detail.Comment: 14 pages, 4 figures, misprints and figures correcte
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