Variational Integrators for Lagrangian Dynamics on Homogeneous Spaces
Melvin Leok
Purdue University
Coauthors: Taeyoung Lee, and N. Harris McClamroch

Variational integrators for Lagrangian flows on a homogeneous space
are constructed by lifting Hamilton's principle from the homogeneous
space to a constrained variational principle on the corresponding Lie
group. This is achieved through a horizontal lift with respect to a
connection that is complementary to the local isotropy subgroup. While
the trajectory on the Lie group depends on the choice of a connection,
the resulting trajectory on the homogeneous space is independent of
the choice of connection.

This approach is applied to Lagrangian flows on two-spheres, where the
corresponding Lie group is SO(3). This approach yields compact
expressions for the continuous and discrete dynamics of mechanisms
consisting of particles with inter-particle length constraints. In
addition, such techniques, when combined with noncommutative harmonic
analysis techniques, provide the basis for constructing geometrically
exact numerical schemes for representing flexible structures and
surfaces arising in modern engineering applications.

This research is partially supported by the National Science Foundation through DMS-0714223 and DMS-0714223.