Abstract
A generic nonintegrable Hamiltonian system is characterized by a critical energy above which chaotic motion sets in. A general method of finding a lower bound to this critical energy that does not require a solution of the equations of motion is discussed. Below the critical energy, the motion is regular everywhere on the energy surface and no instabilities can develop. The method is applied to a practical situation encountered in modern large accelerators, where the transverse motion of the particles in an arrangement of quadrupole, sextupole, and octupole magnetic elements may become chaotic. The chaotic dynamical aperture of the beam is calculated as a function of a dimensionless strength parameter. The estimated critical energy is compared with that obtained from detailed studies of the Poincar\'e sections of the above system at various energies.
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