We study all the symmetries of the free Schrodinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrodinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.
In this paper we study the Darboux transformations of planar vector fields
of Schr odinger type. Using the isogaloisian property of Darboux transformation we prove
the \invariance" of the objects of the \Darboux theory of integrability". In particular,
we also show how the shape invariance property of the potential is important in order to
preserve the structure of the transformed vector field. Finally, as illustration of these results,
some examples of planar vector fields coming from supersymmetric quantum mechanics are
This review paper is devoted to presenting the standard multisymplectic formulation
for describing geometrically classical field theories, both the regular and singular cases.
First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler–Lagrange and the Hamilton–De Donder–Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.