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Energy balance invariance for interacting particle systems

Yavari, Arash and Marsden, Jerrold E. (2009) Energy balance invariance for interacting particle systems. Zeitschrift für Angewandte Mathematik und Mechanik, 60 (4). pp. 723-738. ISSN 0044-2275.

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This paper studies the principle of invariance of balance of energy and its consequences for a system of interacting particles under groups of transformations. Balance of energy and its invariance is first examined in Euclidean space. Unlike the case of continuous media, it is shown that conservation and balance laws do not follow from the assumption of invariance of balance of energy under time-dependent isometries of the ambient space. However, the postulate of invariance of balance of energy under arbitrary diffeomorphisms of the ambient (Euclidean) space, does yield the correct conservation and balance laws. These ideas are then extended to the case when the ambient space is a Riemannian manifold. Pairwise interactions in the case of geodesically complete Riemannian ambient manifolds are defined by assuming that the interaction potential explicitly depends on the pairwise distances of particles. Postulating balance of energy and its invariance under arbitrary time-dependent spatial diffeomorphisms yields balance of linear momentum. It is seen that pairwise forces are directed along tangents to geodesics at their end points. One also obtains a discrete version of the Doyle–Ericksen formula, which relates the magnitude of internal forces to the rate of change of the interatomic energy with respect to a discrete metric that is related to the background metric.

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Additional Information:© 2009 Springer. Received: 29 May 2008. Published online: 10 October 2008. Dedicated to the memory of Shahram Kavianpour (1975-2007). Jerrold E. Marsden: Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Arash Yavari: Research supported by the Georgia Institute of Technology.
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Subject Keywords:Continuum mechanics; particle mechanics; energy balance; covariance
Record Number:CaltechAUTHORS:20090904-135011987
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:15627
Deposited By: George Porter
Deposited On:23 Sep 2009 19:03
Last Modified:26 Dec 2012 11:19

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