Perturbation Theory for Hamilton's Principal Function

Abstract

OLUTIONStotwo-pointboundaryvalueproblems(2BVPs)are of significant importance in the field of astrodynamics and have been subject to extensive research over the years. However, the lack of procedures that automatically converge to the desired solution remains as a fundamental difficulty in solving these problems. Usually, solutions involve open-ended iterative methods that often have no guaranteed convergence and require a good initial guess. Examples of these methods include the method of homotopy, multipleshootingcombinedwithNewton’siteration,andavarietyof other techniques [1]. Evenformanyproblemswithknownsolutions,implicitequations must be solved iteratively to satisfy the 2BVP. In the two-body problem, this situation occurs in finding a solution to Lambert’s problem, which involves iteration [2,3]. Guibout and Scheeres outlined a novel approach for solving 2BVPs usinggenerating functionsfor canonicaltransformations [4]. Their approach requires the system to be a Hamiltonian dynamical system and relies on solving the Hamilton–Jacobi equation, which has its base in Hamilton’s principle. When a perturbation is present, the solution to the nominal problem isno longer valid (althoughit canbe used as an initial guess to an iterative method), and one must resort to numerical methods. Conventional solutions to 2BVPs involve numerical integration schemesandopen-endediterativesolutions.Thesetypesofsolutions focus on finding an optimal transfer cost around the vicinity of the nominalsolution.Moreover,withsuchanapproach,onecouldeasily find the needed correction to the initial impulse in order to hit the desired target. However, this technique will not portray an accurate description of the system and its behavior. It will concentrate on one target point and find the best solution. While in practice this is desirable, it does not obtain a detailed analysis of the vicinity of the desired solution. Guibout and Scheeres [4] also suggested using Hamilton’s principal function (HPF) to solve the 2BVP for the two-body problem. This function is derived directly from Hamilton’s principle and yields solutions to the equations of motion through “simple differentiations and eliminations” [5]. Although the generating function of the canonical transformation and HPF have different physical significance, they are intimately related. HPF allows the initial and final endpoints and times to be dynamic; that is, they are free to change without affecting the structure of the function. Generating functions, on the other hand, has static initial conditions, viewed as constants of motion, with only the final endpoints being dynamic. In this paper, an analytical perturbation technique is developed, and it solves the 2BVP of a perturbed system using HPF. This technique finds its applications in the two-body problem, configuration of spacecraft formations, optimal control problems, and a variety of problems in astrodynamics and other areas as well. This theory is applied to the problem of two bodies, and it can be used to obtain the solution to the system under perturbing forces. For the nominal two-body 2BVP, Lambert’s problem can be solved iteratively in order to obtain desired solutions; therefore, the solution to these types of problems is well established and generally straightforward. Using analytical expressions developed by the perturbation theory developed in this paper allows one to obtain a closed-form solution for the perturbed Lambert’s problem. Therefore, a family of solutions in the vicinity of the desired target can be obtained. This fact helps us better understand the dynamics of the system, even when perturbing forces are present. Solutions to the perturbed system are obtained by analytically expanding HPF around the nominal solution using a small parameter.