Application of the fundamental equation to celestial mechanics and astrodynamics

This paper proposes a new general approach for describing, generating and controlling the trajectory of an object by combining recent advances in analytical dynamics with the underlying theorems and concepts from differential geometry. By using the geometric construct of curvature to define an object's motion and applying the fundamental equation of constrained dynamics, the resulting solutions are both explicit and exact for the minimum acceleration necessary to maintain the specified trajectory. The equations detailing the control force required to follow the selected trajectory can be expressed in closed form, regardless if the object is in Keplerian free-flight about a single central body or following a non-Keplerian trajectory in a highly disturbed environment. Examples are provided in both the inertial and non-inertial frames to demonstrate the utility of this combined approach for solving common problems in both celestial mechanics and astrodynamics. The practical aspects of exploiting curvature for maneuver and mission planning is also investigated resulting in the formulation of the Generalized Transfer Equation which extends the method of patched conics to include any curve.