Existence of stationary geodesics of left-invariant Lagrangians (Q2716822)

In a previous work [\textit{J. Szenthe}, Univ. Iagel. Acta Math. 38, 99-103 (2000)] the author proved the existence of stationary geodesics of left-invariant Riemannian metrics [see also \textit{V. Arnold}, Ann. Inst. Fourier 16, No. 1, 319-361 (1966; Zbl 0148.45301)]. In this paper the author generalizes this result and shows the existence of stationary geodesics for regular Lagrangians which are left-invariant under the action of compact and connected Lie groups, and such that they are first integrals of their associated Lagrangian vector fields. The existence of infinitely many stationary geodesics is established for any compact semi-simple Lie group of rank \(\geq 2\). The main results can be stated as follows. NEWLINENEWLINENEWLINETheorem 2.3: Let \(G\) be a compact Lie group and \(L:TG\to{\mathbb R}\) a left-invariant Lagrangian which is a first integral of its Lagrangian field. Then \(v\in T_eG-\{0_e\}\) is a geodesic vector if and only if \(v\) is the image under the equivariant immersion \(\chi:G/G_v\to G(v)\subset T_eG\) of a critical point of the restricted Lagrangian \(L\circ\chi:G/G_v\to{\mathbb R}\). NEWLINENEWLINENEWLINETheorem 3.2: Let \(G\) be a compact connected Lie group and \(L:TG\to{\mathbb R}\) a left-invariant Lagrangian which is a first integral of its Lagrangian field. Then \(L\) has at least one stationary geodesic. If, in particular, \(G\) is also semi-simple and of rank \(\geq 2\) then \(L\) has infinitely many stationary geodesics.