Decoupled Phase Space Flexing in Autonomous Hamiltonian Systems

Hamiltonian dynamical systems can be studied with the tools of symplectic geometry, which constrain how volumes of space deform. A better understanding of phase space flexing outlines how control strategies can leverage its natural deformation to reduce control effort. This letter establishes analytical forecasts for the stretching of phase space in the vicinity of a reference trajectory. For autonomous Hamiltonian systems, we leverage symplectic invariants and exterior algebraic forms to predict the expansion and contraction of phase space along both the local flow direction and the local gradient direction. This isolates an area-preserving sub-pocket of phase space spanned by these two directions. The stretching results are demonstrated numerically and linked to control effort in the two-body problem.