This paper provides a modern presentation of Noether's theory in the realm of classical dynamics, with application to the problem of a particle submitted to both a potential and a linear dissipation. After a review of the close relationships between Noether symmetries and first integrals, we investigate the variational point symmetries of the Lagrangian introduced by Bateman, Caldirola and Kanai. This analysis leads to the determination of all the time-independent potentials allowing such symmetries, in the one-dimensional and the radial cases. Then we develop a symmetry-based transformation of Lagrangians into autonomous others, and apply it to our problem. To be complete, we enlarge the study to Lie point symmetries which we associate logically to Noether ones. Finally, we succinctly address the issue of a 'weakened' Noether's theory, in connection with 'on-flows' symmetries and non-local constant of motions, for it has a direct physical interpretation in our specific problem. Since the Lagrangian we use gives rise to simple calculations, we hope that this work will be of didactic interest to graduate students, and give teaching material as well as food for thought for physicists regarding Noether's theory and the recent developments around the idea of symmetry in classical mechanics. Lagrangian ‡ The BCK Lagrangian can be adapted to a time-dependent dissipation rate; it suffices to replace γt by γ(t) dt in the exponential factor of (1).
Unique magnetic ordering has been observed in ͑110͒Eu epitaxial films at low temperature: In contrast to bulk Eu, the magnetic helix propagating along the in-plane ͓001͔ direction vanishes on cooling below a temperature T d , at the benefit of the other two. In addition, the two remaining propagation vectors continuously rotate towards the ͓110͔ growth direction. Both the temperature T d and the rotation of wave vectors exhibit a pronounced dependence on film thickness, as a consequence of a thickness-and temperature-dependent lattice clamping effect that distorts the Eu lattice at low temperature.
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