4 research outputs found
Cubic polynomials on Lie groups: reduction of the Hamiltonian system
This paper analyzes the optimal control problem of cubic polynomials on
compact Lie groups from a Hamiltonian point of view and its symmetries. The
dynamics of the problem is described by a presymplectic formalism associated
with the canonical symplectic form on the cotangent bundle of the semidirect
product of the Lie group and its Lie algebra. Using these control geometric
tools, the relation between the Hamiltonian approach developed here and the
known variational one is analyzed. After making explicit the left trivialized
system, we use the technique of Marsden-Weinstein reduction to remove the
symmetries of the Hamiltonian system. In view of the reduced dynamics, we are
able to guarantee, by means of the Lie-Cartan theorem, the existence of a
considerable number of independent integrals of motion in involution.Comment: 20 pages. Final version which incorporates the Corrigendum recently
published (J. Phys. A: Math. Theor. 46 189501, 2013
A general framework for quantum splines
Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on u∗(n) which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.The work of L. Abrunheiro was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (“FCT–Fundaçao para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013. The work of M. Camarinha and P. Santos was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. The work of J. Clemente-Gallardo and J. C. Cuchí was partially covered by MICINN grants FIS2013-46159-C3-2-P and MTM2012-33575 and by DGA Grant 2016-24/1
Cubic polynomials and optimal control on compact Lie groups
This paper analyzes the Riemannian cubic polynomials’s problem from
a Hamiltonian point of view. The description of the problem on compact Lie groups
is particulary explored. The state space of the second order optimal control problem
considered is the tangent bundle of the Lie group which also has a group structure.
The dynamics of the problem is described by a presymplectic formalism associated
with the canonical symplectic form on the cotangent bundle of the tangent bundle.
Using these control geometrical tools, the equivalence between the Hamiltonian
approach developed here and the known variational one is verified. Moreover, the
equivalence allows us to deduce two invariants along the cubic polynomials which
are in involution