Kuzmak A. Brachistochrone problem in classical and quantum mechanics

We solved the brachistochrone problem in the Schwarzschild metric. Also we study the quantum brachistochrone evolution for a system of two spins described by an anisotropic Heisenberg Hamiltonian. A two-step method for the preparation of an arbitrary quantum state of two qubits in the form of the Schmidt decomposition is proposed. The quantum evolution of two-spin system represented by the isotropic Heisenberg Hamiltonian couplings in the magnetic field directed along the z-axis was studied. We conclude that this evolution happens on the manifold which is a torus. Finally, we consider quantum brachistochrone evolution for an arbitrary spin system on rotational manifolds.