Pattern Evocation and Energy-Momentum Integration of the Double Spherical Pendulum

This thesis explores pattern evocation and energy-momentum integration of the double spherical pendulum. Pattern evocation is a phenomenon where patterns emerge when the ow of a dynamical system is viewed in a frame that rotates relative to the inertial frame. Energy-momentum integration integrates the equations of motion and exactly preserves energy and momentum at each time step. The thesis begins with a summary of the theory on pattern evocation for Hamiltonian systems with symmetry. The result of this theory is that if the motion in the reduced space is periodic, quasiperiodic, or almost periodic, respectively, then in a suitably chosen rotating frame with constant velocity, the motion in the unreduced space is also periodic, quasiperiodic, or almost periodic, respectively. The motion in this rotating frame may have a particular pattern or symmetry. Examples of this theory are demonstrated for the double spherical pendulum. A di erential-algebraic model is created for the double spherical pendulum and is integrated with a publically available simulation package called MEXX. This simulation technique is described followed by a description of an energy-momentum integrator. The thesis concludes with a comparison of the energy-momentum integrator and the MEXX simulation.