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We optimize three-dimensional snake kinematics for locomotor efficiency. We assume a general space-curve representation of the snake backbone with small-to-moderate lifting off the ground and negligible body inertia. The cost of locomotion includes work against friction and internal viscous dissipation. When restricted to planar kinematics, our population-based optimization method finds the same types of optima as a previous Newton-based method. With lifting, a few types of optimal motions prevail. We have an s-shaped body with alternating lifting of the middle and ends at small-to-moderate transverse friction. With large transverse friction, curling and sliding motions are typical at small viscous dissipation, replaced by large-amplitude bending at large viscous dissipation. With small viscous dissipation, we find local optima that resemble sidewinding motions across friction coefficient space. They are always suboptimal to alternating lifting motions, with average input power 10–100% higher. 

We study the deformations of elastic filaments confined within slowly shrinking circular boundaries, under contact forces with friction. We perform computations with a spring-lattice model that deforms like a thin inextensible filament of uniform bending stiffness. Early in the deformation, two lobes of the filament make contact. If the friction coefficient is small enough, one lobe slides inside the other; otherwise, the lobes move together or one lobe bifurcates the other. There follows a sequence of deformations that is a mixture of spiralling and bifurcations, primarily the former with small friction and the latter with large friction. With zero friction, a simple model predicts that the maximum curvature and the total elastic energy scale as the wall radius to the − 3 / 2 and − 2 powers, respectively. With non-zero friction, the elastic energy follows a similar scaling but with a prefactor up to eight times larger, due to delayering and bending with a range of small curvatures. For friction coefficients as large as 1, the deformations are qualitatively similar with and without friction at the outer wall. Above 1, the wall friction case becomes dominated by buckling near the wall.