Warren Weckesser Institute for Mathematics and its Applications 400 Lind Hall 207 Church Street SE University of Minnesota Minneapolis, MN 55455-0436 (612) 624-6066 Fax: (612) 626-7370 Email: weck@ima.umn.edu

Informal Seminar on Bifurcation with Symmetry

Selected Publications

• Stabilization of the inverted linearized pendulum by high frequency vibrations, with Mark Levi.
  SIAM Review, 37(2), pp. 219-223, June 1995.
  We give a simple geometric explanation for why an inverted pendulum becomes stable when the pivot point undergoes a high frequency vibration.
  
  
• A ball rolling on a freely spinning turntable.
  American Journal of Physics, 65(8), pp. 736-738, August 1997.
  I show that the path of a ball rolling on a freely spinning turntable is a conic section. This generalizes the ``well known'' fact that the path of a ball rolling on a turntable that spins with a constant angular velocity is a circle.
  

Preprints

• Stability of the relative equilibria in a class of mechanical systems with rotational symmetry.
  We consider mechanical systems that are invariant under the action of the group SO(2). In particular, we consider systems that possess an invariant equilibrium at which the quadratic part of the potential energy is positive definite. We show that, generically, all relative equilibria sufficiently close to the equilibrium are linearly orbitally stable. In addition, we show that the relative equilibria in the first family are nonlinearly orbitally stable, and that the standard variational method for proving nonlinear orbital stability fails in all families after the first. We apply these results to the double spherical pendulum and to a model problem with nine degrees of freedom.
  
  
• More soon...

Other Stuff