Chaos and nonunitary evolution in nonintegrable Hamiltonian systems

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Abstract
The Hamilton-Jacobi canonical transformation theory is extended to treat nonintegrable Hamiltonian systems with a continuous Fourier spectrum. By a natural analytic continuation of the Fourier variable to the complex plane, a nonunitary operator for the canonical transformation is obtained. We apply this transformation to describe Chirikov’s diffusion process near a separatrix in nonlinear systems with two degrees of freedom. Our formalism shows a clear distinction between the irreversible evolution of an ensemble with a finite measure from the reversible evolution of a trajectory in nonintegrable systems with chaotic motion. The condition for obtaining the irreversible kinetic equation in Hamiltonian systems is connected to the condition for the existence of homoclinic points around the separatrix. We also show that Prigogine’s dissipativity condition in nonequilibrium statistical systems is equivalent to the nonintegrability condition for nonlinear systems with a few degrees of freedom.