Swimming in Curved Surfaces and Gauss Curvature

• Main Authors: Leonardo Solanilla, William O Clavijo, Yessica P Velasco
• Format: Article
• Language: English
• Published: Pontificia Universidad Javeriana 2018-08-01
• Series: Universitas Scientiarum
• Subjects: Non-Euclidean Differential Geometry; Local Riemannian Geometry; Lagrangian Formalism; Equations of Motion
• Online Access: https://revistas.javeriana.edu.co/index.php/scientarium/article/view/20893
• ISSN: 0122-7483; 2027-1352

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 translationis some what directly proportional to the Gauss curvature of the surface at the point where the lizardlies.