Harmonic Maps, Conservation Laws and Moving Frames

2002 ◽  
Author(s):  
Frédéric Hélein
Keyword(s):  
Conservation Laws ◽  
Harmonic Maps ◽  
Moving Frames

scholarly journals Moving Frames and Conservation Laws for Euclidean Invariant Lagrangians

2012 ◽  
Vol 130 (2) ◽  
pp. 134-166 ◽  
Author(s):  
T. M. N. Gonçalves ◽  
E. L. Mansfield
Keyword(s):  
Conservation Laws ◽  
Moving Frames

scholarly journals Moving frames and Noether’s finite difference conservation laws I

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
E L Mansfield ◽  
A Rojo-Echeburúa ◽  
P E Hydon ◽  
L Peng
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Group Action ◽  
Lie Group ◽  
Difference Equations ◽  
Continuum Limit ◽  
Moving Frame ◽  
Moving Frames ◽  
Lagrange Equations ◽  
The Difference

Abstract We consider the calculation of Euler–Lagrange systems of ordinary difference equations, including the difference Noether’s theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We introduce the difference moving frame, a natural discrete moving frame that is adapted to difference equations by prolongation conditions. For any Lagrangian that is invariant under a Lie group action on the space of dependent variables, we show that 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 a difference moving frame and the invariants. We show that this form of the laws can significantly ease the problem of solving the Euler–Lagrange equations, and we also show how to use a difference frame to integrate Lie group invariant difference equations. In this Part I, we illustrate the theory by applications to Lagrangians invariant under various solvable Lie groups. The theory is also generalized to deal with variational symmetries that do not leave the Lagrangian invariant. 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. We show the calculations for a discretization of the Lagrangian for Euler’s elastica, and compare our discrete solution to that of its smooth continuum limit.

scholarly journals MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE

2016 ◽  
Vol 4 ◽  
Author(s):  
TÂNIA M. N. GONÇALVES ◽  
ELIZABETH L. MANSFIELD
Keyword(s):  
Conservation Laws ◽  
Group Action ◽  
Lie Group ◽  
Moving Frame ◽  
Moving Frames ◽  
Lagrange Equations ◽  
Linear Action ◽  
Independent Variables ◽  
Lie Group Action ◽  
Standard Linear

In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.

scholarly journals On Moving Frames and Noether’s Conservation Laws

2011 ◽  
Vol 128 (1) ◽  
pp. 1-29 ◽  
Author(s):  
T. M. N. Gonçalves ◽  
E. L. Mansfield
Keyword(s):  
Conservation Laws ◽  
Moving Frames

scholarly journals Moving frames and Noether’s finite difference conservation laws II

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
E L Mansfield ◽  
A Rojo-Echeburúa
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Lie Group ◽  
Difference Equations ◽  
Adjoint Action ◽  
Moving Frame ◽  
Integration Step ◽  
Moving Frames ◽  
Lagrange Equations ◽  
Common Structure

Abstract In this second part of the paper, we consider finite difference Lagrangians that are invariant under linear and projective actions of $SL(2)$, and the linear equi-affine action that preserves area in the plane. We first find the generating invariants, and then use the results of the first part of the paper to write the Euler–Lagrange difference equations and Noether’s difference conservation laws for any invariant Lagrangian, in terms of the invariants and a difference moving frame. We then give the details of the final integration step, assuming the Euler Lagrange equations have been solved for the invariants. This last step relies on understanding the adjoint action of the Lie group on its Lie algebra. We also use methods to integrate Lie group invariant difference equations developed in Part I. Effectively, for all three actions, we show that solutions to the Euler–Lagrange equations, in terms of the original dependent variables, share a common structure for the whole set of Lagrangians invariant under each given group action, once the invariants are known as functions on the lattice.

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