  We examine the problem of snake-like locomotion by studying a system
consisting of a planar inextensible elastic rod that is able to control its
spontaneous curvature. Using a Cosserat model we derive, through variational
principles, the equations of motion for two special cases: one in which the
system is confined inside a frictionless channel, and one in which it is placed
in an anisotropic frictional environment, modeling the dynamical setting of the
slithering of snakes on flat surfaces. The presence of constraints in both
cases leads to non-standard boundary conditions, that allow us to close the
equations of motion reducing them to a differential and an integro-differential
equation, respectively, for one end point (the tail) of the active rod. For the
snake-like case we also provide analytic solutions for a special class of
motions. We highlight the role of the spontaneous curvature in the pushing (and
the steering, in the snake-like setting) needed to power locomotion.
Comparisons with available experiments confirm that the model is able to
capture many of the essential findings in the zoological literature. The
complete solvability and the existence of analytic solutions offers a tool that
may prove valuable for the design of bio-inspired soft robots.
