Keywords
Local Riemannian Geometry
Lagrangian Formalism
Equations of Motion.
How to Cite
Abstract
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.
Atkins R (2011)
The Lie Algebra of Local Killing Fields.
The Open Mathematics Journal 4 5-11
Avron JE, Kenneth O (2006)
Swimming in curved space or the Baron and the cat.
New Journal of Physics 8 68 1-15
Cherman A, Delgado J, Duda F, Ehlers K, Koiller J, Montgomery R (2000)
Low Reynolds Number Swimming in Two Dimensions.
Proceedings III International Symposium Hamiltonian Systems and Celestial Mechanics
eds Delgado J, Lacomba EA, Pérez-Chavela E, Llibre J (World Scientific Publishing) pp 32-62
Cuellar DA (2015)
Geometría del movimiento de cuerpos deformables en superficies curvas.
Universidad del Tolima Trabajo de grado
Littlejohn RG, Reinsch M (1997)
Gauge fields in the separation of rotations and internal motions in the n-body problem.
Reviews of Modern Physics 59 213-275
Shapere A, Wilczek F (1989)
Geometry of self-propulsion at low Reynolds number.
Journal of Fluid Mechanics 198 557-585
Stoker JJ (1969)
Differential Geometry.
(New York: Wiley)
Wisdom J (2003)
Swimming in Spacetime: Motion by Cyclic Changes in Body Shape.
Science 299 1865-1869
Univ. Sci. is registered under a Creative Commons Attribution 4.0 International Public License. Thus, this work may be reproduced, distributed, and publicly shared in digital format, as long as the names of the authors and Pontificia Universidad Javeriana are acknowledged. Others are allowed to quote, adapt, transform, auto-archive, republish, and create based on this material, for any purpose (even commercial ones), provided the authorship is duly acknowledged, a link to the original work is provided, and it is specified if changes have been made. Pontificia Universidad Javeriana does not hold the rights of published works and the authors are solely responsible for the contents of their works; they keep the moral, intellectual, privacy, and publicity rights. Approving the intervention of the work (review, copy-editing, translation, layout) and the following outreach, are granted through an use license and not through an assignment of rights. This means the journal and Pontificia Universidad Javeriana cannot be held responsible for any ethical malpractice by the authors. As a consequence of the protection granted by the use license, the journal is not required to publish recantations or modify information already published, unless the errata stems from the editorial management process. Publishing contents in this journal does not generate royalties for contributors.