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.
"Moving frames and conservation laws of a Lagrangian invariant under the Hyperbolic Rotation-Translation group." Hokkaido Math. J. 47 (3) 557 - 579, October 2018. https://doi.org/10.14492/hokmj/1537948831