Before revision:  This paper presents a comprehensive theory of optimization fabrics which are second-order differential equations that encode nominal behaviors on a space and are guaranteed to optimize when forced by a potential function.
After revision:  This paper presents a comprehensive theoretical framework expanding this approach showing that there is a much larger class of differential equations generalizing classical mechanical systems (and the broader class of Lagrangian systems) and greatly expanding their expressivity while maintaining the same governing stability principles. At the core of our framework is a class of differential equations we call fabrics which constitute a behavioral medium across which we can optimize a potential function.