Abstract

SOLANILLA, Leonardo; CLAVIJO, William O.  and  VELASCO, Yessica P.. Swimming in Curved Surfaces and Gauss Curvature. Univ. Sci. [online]. 2018, vol.23, n.2, pp.319-331. ISSN 0122-7483.  https://doi.org/10.11144/javeriana.sc23-2.sics.

The Cartesian-Newtonian paradigm of mechanics establishes that, within an inertial frame, a body either remains at rest or moves uniformly on a line, unless a force acts externally upon it. This crucial assertion breaks down when the classical concepts of space, time and measurement reveal to be inadequate. If, for example, the space is non-flat, an effective translation might occur from rest in the absence of external applied force. In this paper we examine mathematically the motion of a small object or lizard on an arbitrary curved surface. Particularly, we allow the lizard’s shape to undergo a cyclic deformation due exclusively to internal forces, so that the total linear momentum is conserved. In addition to the fact that the deformation produces a swimming event, we prove -under fairly simplifying assumptions- that such a translation is somewhat directly proportional to the Gauss curvature of the surface at the point where the lizard lies.

Keywords : equations of motion; Lagrangian formalism; local Riemannian geometry; non-Euclidean differential geometry.

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