The geometry of Newton's law and rigid systems

Marco Modugno; Raffaele Vitolo

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 3, page 197-229
  • ISSN: 0044-8753

Abstract

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We start by formulating geometrically the Newton’s law for a classical free particle in terms of Riemannian geometry, as pattern for subsequent developments. For constrained systems we have intrinsic and extrinsic viewpoints, with respect to the environmental space. Multi–particle systems are modelled on n -th products of the pattern model. We apply the above scheme to discrete rigid systems. We study the splitting of the tangent and cotangent environmental space into the three components of center of mass, of relative velocities and of the orthogonal subspace. This splitting yields the classical components of linear and angular momentum (which here arise from a purely geometric construction) and, moreover, a third non standard component. The third projection yields a new explicit formula for the reaction force in the nodes of the rigid constraint.

How to cite

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Modugno, Marco, and Vitolo, Raffaele. "The geometry of Newton's law and rigid systems." Archivum Mathematicum 043.3 (2007): 197-229. <http://eudml.org/doc/250175>.

@article{Modugno2007,
abstract = {We start by formulating geometrically the Newton’s law for a classical free particle in terms of Riemannian geometry, as pattern for subsequent developments. For constrained systems we have intrinsic and extrinsic viewpoints, with respect to the environmental space. Multi–particle systems are modelled on $n$-th products of the pattern model. We apply the above scheme to discrete rigid systems. We study the splitting of the tangent and cotangent environmental space into the three components of center of mass, of relative velocities and of the orthogonal subspace. This splitting yields the classical components of linear and angular momentum (which here arise from a purely geometric construction) and, moreover, a third non standard component. The third projection yields a new explicit formula for the reaction force in the nodes of the rigid constraint.},
author = {Modugno, Marco, Vitolo, Raffaele},
journal = {Archivum Mathematicum},
keywords = {classical mechanics; rigid system; Newton’s law; Riemannian geometry; classical mechanics; rigid system; Newton's law; Riemannian geometry},
language = {eng},
number = {3},
pages = {197-229},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The geometry of Newton's law and rigid systems},
url = {http://eudml.org/doc/250175},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Modugno, Marco
AU - Vitolo, Raffaele
TI - The geometry of Newton's law and rigid systems
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 3
SP - 197
EP - 229
AB - We start by formulating geometrically the Newton’s law for a classical free particle in terms of Riemannian geometry, as pattern for subsequent developments. For constrained systems we have intrinsic and extrinsic viewpoints, with respect to the environmental space. Multi–particle systems are modelled on $n$-th products of the pattern model. We apply the above scheme to discrete rigid systems. We study the splitting of the tangent and cotangent environmental space into the three components of center of mass, of relative velocities and of the orthogonal subspace. This splitting yields the classical components of linear and angular momentum (which here arise from a purely geometric construction) and, moreover, a third non standard component. The third projection yields a new explicit formula for the reaction force in the nodes of the rigid constraint.
LA - eng
KW - classical mechanics; rigid system; Newton’s law; Riemannian geometry; classical mechanics; rigid system; Newton's law; Riemannian geometry
UR - http://eudml.org/doc/250175
ER -

References

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