Abstract
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.
Highlights
Noether’s First Theorem states that for systems coming from a variational principle, conservation laws may be obtained from Lie group actions which leave the Lagrangian invariant
We move on to the invariant calculus of variations; we show how the invariantized Euler–Lagrange equations are obtained in a way similar to that of the Euler–Lagrange equations in the original variables
In [8, Theorem 3], it was shown that for Lagrangians which are invariant under a certain group action, and whose independent variables are left unchanged by that action, the conservation laws can be written as the product of the Adjoint representation of a moving frame for the Lie group action and vectors of invariants; in this new format, the laws are handled and analysed more
Summary
Noether’s First Theorem states that for systems coming from a variational principle, conservation laws may be obtained from Lie group actions which leave the Lagrangian invariant. In [8, 17], for the case where the invariant Lagrangians may be parametrized so that the independent variables are each invariant under the group action, the authors were able to calculate the invariantized Euler–Lagrange system in terms of the standard Euler operator and a ‘syzygy’ operator specific to the action They obtained the linear space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame for the Lie group action. This new structure for the conservation laws allowed the calculations for the extremals to be reduced and given in the original variables, once the Euler–Lagrange system was solved for the invariants.
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