Brake orbits and magnetic twistings in two degrees of freedom Hamiltonian dynamical systems

We develop a theory of brake orbits in the context of a two degrees of freedom Hamiltonian dynamical system whose configuration manifold is a nonempty open submanifold of the real plane, whose Hamiltonian is of the mechanical kinetic minus potential type, and whose underlying symplectic structure is the canonical one modified by a magnetic twisting. A brake orbit is one in which the motion has at least two instances of zero momentum or brakes. Only in the absence of a twisting is every brake orbit guaranteed to extend to a periodic brake orbit. We introduce the brake equation which implicitly encodes all the information about brake orbits. We develop two notions of nondegeneracy for a brake orbit, one with respect to interbraking time (the time between two brakes), and one with respect to energy. We develop a termination principle and investigate the energy-interbraking time relationship for one-parameter families of brake orbits. We prove a local existence theorem which states that in a neighbourhood of a saddle-center (an equilibrium whose eigenvalues are {dollar}{lcub}pm{rcub}lambda, {lcub}pm{rcub}iomega{dollar} where {dollar}lambda{dollar} and {dollar}omega{dollar} are real and positive) there exist infinitely many one parameter families of nondegenerate (with respect to energy) brake orbits terminating at the saddle-center whenever the twisting at the saddle-center is nonzero. The proof of this is based on Poincare's continuation method. We prove conditional existence theorems for the existence of brake orbits of prescribed energy and the existence of heteroclinic orbits connecting two hyperbolic equilibria. The proofs of these are based on variational analysis and the Hartman-Grobman Theorem. These two conditional existence theorems are supporting evidence of a brake orbit method for proving the existence of orbits homoclinic and heteroclinic to hyberbolic equilibria.