Lambert's problem in orbital dynamics: a self--contained introduction

TL;DR

This paper provides a clear, comprehensive introduction to Lambert's problem in orbital mechanics, focusing on elliptical trajectories and minimal background assumptions, serving as an accessible resource for students and researchers.

Contribution

It offers a unified, didactic derivation of Lambert's problem, emphasizing simplicity and clarity for educational purposes.

Findings

Derivation of Lambert's problem for elliptical trajectories

Focus on minimal physics and mathematics background

Provides a comprehensive reference for quick understanding

Abstract

Lambert's problem is a classical boundary value problem in analytical mechanics. It arises when trying to determine the energy required to place a particle, subject to a central gravitational potential, in a "free fall" trajectory connecting two given points on a desired travel time. Due to its mathematical beauty and its relevance in aerospace engineering, it has been and remains the object of attention of countless engineers, mathematicians (pure and applied), and physicists seeking to produce efficient solution algorithms. In this expository article, didactic in nature, we present a unified and comprehensive derivation that assumes only a minimal background in physics and mathematics. We focus on the simplest unperturbed case and carefully develop the argument for elliptical trajectories. The goal is to provide a single reference that can serve as an accelerated introduction for students and researchers interested in a quick introduction to the subject.