The Lie algebra \(\mathfrak{sl}(2,\mathbb{R})\) and Noether point symmetries of Lagrangian systems (Q2418904)

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].