A principle of ‘Least Effort’ to describe the natural movements of animals

This paper presents a general theory for the dynamical properties of natural movements of animals with articulated bodies. The theory assumes only that the animal’s choice of movement is in some way optimal, whilst respecting classical laws of physics. This leads to a variational principle governing the movement of animals, represented by a simple equation, the solution of which is life-like motion. Although many ad hoc algorithms exist for animating creatures, the present paper brings animal movement into the realm of mathematical physics by formalising life-like dynamics in an explicit, physically motivated equation with well defined solutions. The variational principle generates graceful movements between chosen beginning and end states without relying on data sets or human intervention to find in-between postures. Importantly, the movements respect the laws of Newtonian dynamics, while also capturing the lazy appearance of the voluntary movements of animals that have evolved to move efficiently. The formalism makes a distinction between passive and active degrees of freedom. The former respect Hamilton’s principle of least action, while the latter respect a principle of least effort, which is non-trivial to implement due to the coupling between passive and active degrees of freedom. Both numerical and analytical solutions to the problem are developed here. Central to the numerical solution is the careful counting of the appropriate number of constraints for the number of variables used. Both approaches are demonstrated on some minimal models. The aim of the paper is not to compete with state-of-the-art computer animation, but to establish a rigorous approach, with potential applications in sports science, veterinary and human medicine as well as computer-generated animation.