Linear and Integrable Systems

Eyad Hasan Hasan

The Lagrangian with linear acceleration can be considered as a model of singular system. The constrained Hamiltonian systems with linear acceleration are investigated by using the Hamilton-Jacobi method. The Hamilton-Jacobi function is constructed by applying the integrability condition on this function to obtain the path integral quantization. It is shown that the equations of motion can be obtained from the action integral

Miki Wadati, Taro Nagao, Kazuhiro Hikami

Charles G Torre

We present a computation of the coherent state path integral for a generic linear system using ``functional methods'' (as opposed to discrete time approaches). The Gaussian phase space path integral is formally given by a determinant built from a first-order differential operator with coherent state boundary conditions. We show how this determinant can be expressed in terms of the symplectic transformation generated by the (in general, time-dependent) quadratic Hamiltonian for the system. We briefly discuss the conditions under which the coherent state path integral for a linear system actually exists. A necessary---but not sufficient---condition for existence of the path integral is that the symplectic transformation generated by the Hamiltonian is (unitarily) implementable on the Fock space for the system.

Alok Sinha

Tomio Yamakoshi Petrosky, H. H. Hasegawa

Anatol Odzijewicz, E. Wawreniuk

The relationship between classical and quantum three one-mode systems interacting in a non-linear way is described. We investigate the integrability of these systems by using the reduction procedure. The reduced coherent states for the quantum system are constructed. We find the explicit formulas for the reproducing measure for these states. Examples of some applications of the obtained results in non-linear quantum optics are presented.