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Noether’s Theorem, smooth and discrete

PhD candidate Ana Rojo-Echeburúa

In this talk, I will illustrate progress, first in the understanding of the mathematical structure of Noether’s conservation laws for a geometric group action, and second their adaptation to various discrete versions.

For both smooth and discrete cases, I will make use of the modern theory of moving frames and its discrete adaptation respectively. I will illustrate this theory for a very well know Lie group action, the projective SL(2) action and compare the results for the smooth and discrete version. I will show that for any Lagrangian that is invariant under this Lie group action, the Euler Lagrange equations can be calculated directly in terms of the invariants of the group action. Furthermore, Noether's conservation laws can be written in terms of the moving frame and the invariants.

Apart from the study of systems that are inherently discrete, one significant application is to obtain geometric variational integrators that have finite difference approximations of the continuous conservation laws embedded a priori. This is achieved by taking an invariant finite difference Lagrangian in which the discrete invariants have the correct continuum limit to their smooth counterparts. I will show the calculations for a discretisation of the Lagrangian for Euler’s elastica, and compare the discrete solution to that of its smooth continuum limit.