In this paper we develop some group theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation and we discuss some of its integrability conditions from a group theoretical perspective. The nonlinear superposition principle also arises in a simple way.
The basic concepts of factorizable problems in one–dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed. The relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail. By the use of some properties of the Riccati equation the… (More)
We show that the finite difference Bäcklund formula for the Schrödinger Hamiltonians is a particular element of the transformation group on the set of Riccati equations considered by two of us in a previous paper. Then, we give a group theoretical explanation to the problem of Hamiltonians related by a first order differential operator. A generalization of… (More)
Several instances of integrable Riccati equations are analyzed from the geometric perspective of the theory of Lie systems. This provides us a unifying viewpoint for previous approaches.
Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the Hamiltonian function being the reduced energy. We study further the algebraic origin of these Poisson structures, showing that… (More)
We recall the Theorem by Lie and Scheffers concerning the characterization of systems of differential equations admitting a superposition function, i.e. those whose general solution can be written in terms of some particular solutions and constants. Each of these systems is related with a Lie algebra, specified by the own Theorem. We expose some recently… (More)
The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be related with equations on a Lie group and with some connections in fiber bundles. We develop two methods for dealing with… (More)
Electrothermal motion in an aqueous solution arises from the action of an electric field on inhomogeneities in the liquid induced by temperature gradients. The temperature field can be produced by the applied electric field through Joule heating, or caused by external sources, such as strong illumination. Electrothermal flows in microsystems are usually… (More)
Net fluid flow induced by ac potentials applied to arrays of co-planar interdigitated microelectrodes is reported. Two types of microelectrode structures have been employed: arrays of unequal width electrodes subjected to a single ac signal, and arrays of identical electrodes subjected to a traveling-wave potential. A square glass chamber was constructed… (More)
Some simple examples from quantum physics and control theory are used to illustrate the application of the theory of Lie systems. We will show, in particular, that for certain physical models both of the corresponding classical and quantum problems can be treated in a similar way, may be up to the replacement of the involved Lie group by a central extension… (More)