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The Lie algebra \(\mathfrak{sl}(2,\mathbb{R})\) and Noether point symmetries of Lagrangian systems. (English) Zbl 1461.70022

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 20th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 2–7, 2018. Sofia: Avangard Prima; Sofia: Bulgarian Academy of Sciences, Institute of Biophysics and Biomedical Engineering. Geom. Integrability Quantization 20, 99-114 (2019).
The paper under review studies Noether point symmetries of a regular Lagrangian \(L\) of kinetic type, defined by \[L(q,\dot q)=\frac 12 <\dot q, \dot q>,\] where \(<\cdot , \cdot >\) denotes a pseudo-Riemannian metric tensor. The structure of the Lie algebra generated by infinitesimal generators of \(L\) which are not Killing vector fields, and its connection to the associated metric of \(L\), are analysed. It is shown that, when the Lie subalgebra of Noether point symmetries under study is isomorphic \(\mathfrak{sl}(2,\mathbb{R})\), there exists a constant of motion independent of time.
An example of a Lagrangian defined by a metric tensor of a surface \(S\) illustrates the relation between realizations of \(\mathfrak{sl}(2,\mathbb{R})\) as subalgebras of Noether point symmetries and geometric properties of \(S\) described by a null sectional curvature. Another example of a Lagrangian, for which the subalgebra of Noether point symmetries is isomorphic to \(\mathfrak{sl}(2,\mathbb{R})\) but the sectional curvature is not necessarily null, is interpreted as a perturbed Lagrangian of the Lagrangian defined by a flat metric.
For the entire collection see [Zbl 1408.53003].

MSC:

70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
70F17 Inverse problems for systems of particles
17B81 Applications of Lie (super)algebras to physics, etc.
34A26 Geometric methods in ordinary differential equations
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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