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

TL;DR

The paper presents a self-contained introduction to Lambert's problem in Keplerian dynamics, deriving the conic-orbit framework and Kepler's laws with explicit equations such as $r = \frac{p}{1+ e\cos\theta}$ and $p = \frac{L^2}{GMm^2}$. It then formulates Lambert's problem for an elliptic transfer and shows how Lagrange's transfer-time construction reduces the problem to a tractable system in auxiliary angles $\alpha$ and $\beta$, yielding the transfer-time equation $\sqrt{\frac{GM}{a^3}}(t_2-t_1) = 2\pi Q + \alpha - \beta - 2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)$ and relations for $\cos\alpha$, $\cos\beta$ in terms of $r_1,r_2,c$, and $a$. These developments identify the conditions for multiple solutions depending on the travel time and revolutions, and point to modern, robust solvers. The article thus provides a compact, self-contained reference connecting conic geometry to practical elliptic-transfer calculations.

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.

Figures (4)

• Figure 1: Left: Conical sections for different values of the eccentricity $e$. Shades of red represent values of $0<e<1$ resulting in ellipses; the color fades towards one. Shades of blue represent values of $e>1$ resulting in hyperbolae with the color fading towards one. The parabola, plotted in violet, is the limiting case $e=1$. The directrix $L$ is plotted as a dashed line, the focus $F$ is marked by an open circle. Right: Relevant geometric markers for ellipses and hyperbolae.
• Figure 2: Polar description of a point $P$ in terms of the semi--major axis $a$ and eccentricity $e$.
• Figure 3: Left: For a fixed angle $\theta$ the polar basis vectors are orthogonal and point in the direction of growth $(1+\Delta)\boldsymbol r$ (in gray) and $\theta+\Delta\theta$ (in red). Right: The velocity of a particle, $\dot{\boldsymbol r}$, can be decomposed in radial and angular components.
• Figure 4: The area of the parallelogram determined by the vectors $\boldsymbol r$ and $\boldsymbol{dr}$ (shaded) is given by the norm $|\boldsymbol{r\times dr}|$. As the position vector $\boldsymbol r$ undergoes a small displacement $\boldsymbol{dr}$ along the elliptic arc, the area of the infinitesimal elliptic sector it traverses (dark shade) is given by $\tfrac{1}{2}|\boldsymbol{r\times dr}|$. Right: The time required to travel between $\boldsymbol{r_1}$ and $\boldsymbol{r_2}$ is the same as that required to travel between $\boldsymbol{r_2}$ and $\boldsymbol{r_4}$.

Theorems & Definitions (2)

• definition thmcounterdefinition: besant1890GlStOd2016
• remark thmcounterremark: On the sign of $e$ in the focal equation