Canonical Solution of Two-Point Boundary-Value Problems

A canonical method is developed to solve two-point boundary-value problems (TPBVP) for Hamiltonian systems with a main integrable part. The method uses the canonical solution of Hamilton-Jacobi equations by separation of variables and canonical perturbation theory. Taking the perturbations into account, the original TPBVPs are transformed to new TPBVPs expressed in terms of the canonical constants, which are variable and satisfy the canonical perturbation equations. Assuming that the minor perturbation Hamiltonian relatively small, the new TPBVPs are solved by linearization. By updating the canonical constants as the new reference point for linearization, the linearized solving process is iterated to achieve high-order precision. The canonical method is shown more specifically for the relative motion of satellite formation flying, with both nonlinear effect and perturbations considered.