The existence of periodic orbits on the sphere (Q1059916)

Besides the so-called Seifert conjecture, \textit{H. Seifert} in fact proved in his paper [see Math. Z. 51, 197-216 (1948; Zbl 0030.22103)] the existence of a periodic orbit on \(S^{2n-1}\) for a class of Lagrangian systems with n degrees of freedom, and proposed a problem for the existence of \(n\) periodic orbits on \(S^{2n-1}\) for the same class. In this note, the author proves two theorems; one generalizes the above mentioned existence theorem of Seifert, and the other gives an affirmative answer to Seifert's problem mentioned above for a special class of the rotationally symmetric Lagrangian systems.