Classical Hamiltonian Systems with balanced loss and gain

Classical Hamiltonian systems with balanced loss and gain are considered in this review. A generic Hamiltonian formulation for systems with space-dependent balanced loss and gain is discussed. It is shown that the loss-gain terms may be removed completely through appropriate co-ordinate transformations with its effect manifested in modifying the strength of the velocity-mediated coupling. The effect of the Lorentz interaction in improving the stability of classical solutions as well as allowing a possibility of defining the corresponding quantum problem consistently on the real line, instead of within Stokes wedges, is also discussed. Several exactly solvable models based on translational and rotational symmetry are discussed which include coupled cubic oscillators, Landau Hamiltonian etc. The role of ${\mathscr{PT}}$-symmetry on the existence of periodic solution in systems with balanced loss and gain is critically analyzed. A few non-${\mathscr{PT}}$-symmetric Hamiltonian as well as non-Hamiltonian systems with balanced loss and gain, which include mechanical as well as extended system, are shown to admit periodic solutions. An example of Hamiltonian chaos within the framework of a non-${\mathscr{PT}}$-symmetric system of coupled Duffing oscillator with balanced loss-gain and/or positional non-conservative forces is discussed. It is conjectured that a non-${\mathscr{PT}}$-symmetric system with balanced loss-gain and without any velocity mediated interaction may admit periodic solution if the linear part of the equations of motion is necessarily ${\mathscr{PT}}$ symmetric —the nonlinear interaction may or may not be ${\mathscr{PT}}$-symmetric. Further, systems with velocity mediated interaction need not be ${\mathscr{PT}}$-symmetric at all in order to admit periodic solutions. Results related to nonlinear Schrödinger and Dirac equations with balanced loss and gain are mentioned briefly. A class of solvable models of oligomers with balanced loss and gain is presented for the first time along with the previously known results.