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Moving frames and conservation laws of a Lagrangian invariant under the hyperbolic rotation-translation group. (English) Zbl 1415.35192
Summary: Noether’s First Theorem guarantees conservation laws provided that the Lagrangian is invariant under a Lie group action. In this paper, via the concept of Killing vector fields and the Minkowski metric, we first construct an important Lie group, known as Hyperbolic Rotation-Translation group. Then, according to Gonçalves and Mansfield’s method, we 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, for Lagrangians, which are invariant under Hyperbolic Rotation-Translation (or HRT) group action, in the case where the independent variables are not invariant.
MSC:
35L65 Hyperbolic conservation laws
58E30 Variational principles in infinite-dimensional spaces
70S10 Symmetries and conservation laws in mechanics of particles and systems
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58D19 Group actions and symmetry properties
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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