Invariance and the Structure of Matter

Sunday, October 25, 2015 - 10:50am - 11:30am
Richard James (University of Minnesota, Twin Cities)
A thread of Jerry Ericksen’s research that runs through numerous subject areas – nonlinear elasticity, shells, liquid and solid crystals – is the exploitation of invariance of the basic equations of mechanics. This work involved the continuous groups, with a particular focus on the invariance arising from the condition of frame-indifference of the constitutive equations. I recall that in Jerry’s informal seminar around 1978, he raised the question, “What can be done with the discrete groups?” We discussed this on the blackboard for a while, particularly focusing on conditions on derivatives -- a simple special case would be the group {1,-1} with the most simple action, and the associated vanishing of the derivative of an even function at the origin. A big fraction of my work in recent years, inspired in part by this discussion, is the interaction of the discrete groups and the equations of mechanics and electromagnetism. It turns out that a discrete group, operating appropriately on a discrete (atomic) structure, gives results that are analogous to our favorite continuous groups acting on the equations of continuum mechanics. I give some examples from molecular dynamics, the Boltzmann equation and Maxwell’s equations.
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