This paper describes a method based upon the classical calculus of variations for solving directly Chebyshev minimax problems which arise in trajectory optimization. The close relationship between minimax problems and problems with state variable inequality constraints is used to gain insight into the minimax problem, and to define an order for minimax functions. The method is applicable to 1) afl problems in which the first time derivative of the minimax function does not contain control variables explicitly, and 2) all problems with a "flat" maximum (including problems in which the first time derivative of the minimax function contains control variables). The theory is applied to both a simple example and a formulation of the optimal re-entry heating problem whose performance index consists of a minimax term for maximum heating rate, a terminal function for maximum crossrange, and a path integral for minimum total heating. Several shooting and gradient-type numerical algorithms are suggested by the approach.