A Lie Group and Principal Bundle Structure Preserving Framework for Numerical Integration of Flexible Body Dynamics

Abstract We present a novel geometric framework for numerical integration of flexible body dynamics, grounded in Lie group and fiber bundle theories. The configuration space of a flexible body is modeled as a principal fiber bundle, where internal deformations define the base manifold and the overall rigid body poses lie in the fibers. Within this formulation, flexible body motion naturally decomposes into horizontal and vertical components. Building on the concept of a connection form, we introduce nonlinear constraints that ensure consistent evolution and coupling of these components. Computations are performed stepwise in instantaneous inertial frames, analogous to momentarily co-moving inertial frames in relativistic mechanics, thereby preserving the invariant linear form of the governing equations of motion while avoiding spurious inertial effects. The global motion is then reconstructed from the stepwise updates. The resulting numerical scheme respects the Lie algebra of the structure group and preserves the bundle structure of the configuration space at each time step. This framework generalizes existing gauge-theoretic approaches to deformable body kinematics [1, 2], providing a robust, structure-preserving approach for integrating coupled flexible body dynamics.