Aas 05-332 a Historical Introduction to the Covector Mapping Principle

In 1696, Johann Bernoulli solved the brachistochrone problem by an ingenious method of combining Fermat’s principle of minimum time, Snell’s law of refraction and “finite element” discretization. This appears to be the first application of a “direct method.” By taking the limits of these “broken-line solutions,” Bernoulli arrived at an equation for the cycloid. About fifty years later (1744), Euler generalized Bernoulli’s direct method for the general problem of finding optimal curves and derived the now-famous Euler-Lagrange equations. Lagrange’s contribution did not come until 1755 when he (Lagrange) showed that Euler’s result could be arrived at by an alternative route of a new calculus. Lagrange’s ideas superceded the Bernoulli-Euler method and paved the way for a calculus of variations that culminated in the 1930s at the University of Chicago. In the late 1950s, the complexity of these variational equations were dramatically reduced by the landmark results of Bellman and Pontryagin. Their results are connected to Karush’s generalization of Lagrange’s yet-another-idea of “undetermined” multipliers. The simplicity of their equations also came with an amazing bonus of greater generality that engineers could now conceive of applying their results to practical problems. In recognizing that the elegant methods of Bellman and Pontryagin were not scalable to space trajectory optimization, astrodynamicists developed a broad set of computational tools that frequently required deep physical insights to solve real-world mission planning problems. In parallel, mathematicians discovered that the equations of Bellman and Pontryagin were incompatible with the original ideas of Bernoulli and Euler. Since the 1960s, intense research within the mathematical community has lead to the notion of “hidden convexity,” set-valued analysis, geometric integrators and many other mathematical topics that have immediate practical consequences, particularly to simplifying complex mission planning problems. This is the story of the covector mapping principle. When combined with a modern computer, it renders difficult trajectory optimization problems remarkably easy that it is now possible to routinely generate even real-time solutions.