Hamiltonian formulation of systems with balanced loss-gain and exactly solvable models (original) (raw)
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Integrable coupled Liénard-type systems with balanced loss and gain
Annals of Physics
A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of coordinates , the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain coefficient .The resulting equations of motion from the Hamiltonian are a system of coupled Liénard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of m+1 integrals of motion are constructed for a system of 2m particles, which are in involution, implying that two-particle systems are completely integrable. A few exact solutions for both the cases are presented for specific choices of the potential and spacedependent gain/loss coefficients , which include periodic stable solutions. Quantization of the system is discussed with the construction of the integrals of motion for specific choices of the potential and gain-loss coefficients. A few quasi-exactly solvable models admitting bound states in appropriate Stoke wedges are presented.
Journal of Physics A, 2019
The quantization of many-body systems with balanced loss and gain is investigated. Two types of models characterized by either translational invariance or rotational symmetry under rotation in a pseudo-Euclidean space are considered. A partial set of integrals of motion are constructed for each type of model. Specific examples for the translational invariant systems include Calogero-type many-body systems with balanced loss and gain, where each particle is interacting with other particles via four-body inverse-square potential plus pair-wise two-body harmonic terms. A many-body system interacting via short range four-body plus six-body inverse square potential with pair-wise two-body harmonic terms in presence of balanced loss and gain is also considered. In general, the eigen values of these two models contain quantized as well as continuous spectra. A completely quantized spectra and bound states involving all the particles may be obtained by employing box-normalization on the particles having continuous spectra. The normalization of the ground state wave functions in appropriate Stoke wedges is discussed. The exact n-particle correlation functions of these two models are obtained through a mapping of the relevant integrals to known results in random matrix theory. It is shown that a rotationally symmetric system with generic many-body potential does not have entirely real spectra, leading to unstable quantum modes. The eigenvalue problem of a Hamiltonian system with balanced loss and gain and admitting dynamical O(2, 1) symmetry is also considered.
Introduction to Hamiltonian Dynamical Systems and the N-Body Problem
Kenneth R. Meyer Glen R. Hall Dan Offin
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N ov 2 01 8 Integrable coupled Li é nard-type systems with balanced loss and gain
2018
A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a suitable choice of co-ordinates, the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain co-efficient.The resulting equations of motion from the Hamiltonian are a system of coupled Liénard-type differential equations. Partially integrable systems are obtained for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. A total number of m+1 integrals of motion are constructed for a system of 2m particles, which are in involution, implying that two-particle systems are completely integrable. A ...
A paradigm for joined Hamiltonian and dissipative systems
Physica D: Nonlinear Phenomena, 1986
A paradigm for describing dynamical systems that have both Hamiltonian and dissipative parts is presented. Features of generalized Hamiltonian systems and metric systems are combined to produce what are called metriplectic systems. The phase space for metriplectic systems is equipped with a bracket operator that has an antisymmetric Poisson bracket part and a symmetric dissipative part. Flows are obtained by means of this bracket together with a quantity called the generalized free energy, which is composed of an energy and a generalized entropy. The generalized entropy is some function of the Casimir invariants of the Poisson bracket. Two examples are considered: (1) a relaxing free rigid body and (2) a plasma collision operator that can be tailored so that the equilibrium state is an arbitrary monotonic function of the energy.
PT -symmetric rational Calogero model with balanced loss and gain
2017
A two body rational Calogero model with balanced loss and gain is investigated. The system yields a Hamiltonian which is symmetric under the combined operation of parity (P) and time reversal (T ) symmetry. It is shown that the system is integrable and exact, stable classical solutions are obtained for particular ranges of the parameters. The corresponding quantum system admits bound state solutions for exactly the same ranges of the parameters for which the classical solutions are stable. The eigen spectra of the system is presented with a discussion on the normalization of the wave functions in proper Stokes wedges. Finally, the Calogero model with balanced loss and gain is studied classically, when the pair-wise harmonic interaction term is replaced by a common confining harmonic potential. The system admits stable solutions for particular ranges of the parameters. However, the integrability and/or exact solvability of the system is obscure due to the presence of the loss and gai...
$\mathcal{PT}$ 𝒫𝒯 -symmetric rational Calogero model with balanced loss and gain
The European Physical Journal Plus, 2017
A two body rational Calogero model with balanced loss and gain is investigated. The system yields a Hamiltonian which is symmetric under the combined operation of parity (P) and time reversal (T) symmetry. It is shown that the system is integrable and exact, stable classical solutions are obtained for particular ranges of the parameters. The corresponding quantum system admits bound state solutions for exactly the same ranges of the parameters for which the classical solutions are stable. The eigen spectra of the system is presented with a discussion on the normalization of the wave functions in proper Stokes wedges. Finally, the Calogero model with balanced loss and gain is studied classically, when the pair-wise harmonic interaction term is replaced by a common confining harmonic potential. The system admits stable solutions for particular ranges of the parameters. However, the integrability and/or exact solvability of the system is obscure due to the presence of the loss and gain terms. The perturbative solutions are obtained and are compared with the numerical results.
Mathematical Modeling of Constrained Hamiltonian Systems
IFAC Proceedings Volumes
Network modelling of unconstrained energy conserving physical systems leads to an intrinsic generalized Hamiltonian formulation of the dynamics. Constrained energy conserving physical systems are directly modelled as implicit Hamiltonian systems with regard to a generalized Dirac structure on the space of energy variables, which can be rewritten (under a nondegeneracy condition on the internal energy) as an explicit generalized Hamiltonian system on Ihe constrained state space. This is specialized to mechanical systems with kinematic constraints.
A class of coupled KdV systems and their bi-Hamiltonian formulation
Journal of Physics A: Mathematical and General, 1998
A Hamiltonian pair with arbitrary constants is proposed and thus a sort of hereditary operators is resulted. All the corresponding systems of evolution equations possess local bi-Hamiltonian formulation and a special choice of the systems leads to the KdV hierarchy. Illustrative examples are given.
Interconnections of input-output Hamiltonian systems with dissipation
2016 IEEE 55th Conference on Decision and Control (CDC), 2016
Recently, negative imaginary and counterclockwise systems have attracted attention as an interesting class of systems, which is well-motivated by applications. In this paper first the formulation and extension of negative imaginary and counterclockwise systems as (nonlinear) inputoutput Hamiltonian systems with dissipation is summarized. Next it is shown how by considering the time-derivative of the outputs a port-Hamiltonian system is obtained, and how this leads to the consideration of alternate passive outputs for port-Hamiltonian systems. Furthermore, a converse result to positive feedback interconnection of input-output Hamiltonian systems with dissipation is obtained, stating that the positive feedback interconnection of two linear systems is an input-output Hamiltonian system with dissipation if and only if the systems themselves are input-output Hamiltonian systems with dissipation. This implies that the Poisson and resistive structure matrices can be redefined in such a way that the interaction between the two systems only takes place via the coupling term in the Hamiltonian of the interconnected system. Subsequently, it is shown how network interconnection of (possibly nonlinear) input-output Hamiltonian systems with dissipation results in another input-output Hamiltonian system with dissipation, and how this leads to a stability analysis of the interconnected system in terms of the Hamiltonians and output mappings of the systems associated to the vertices, as well as the topology of the network.