SODEs and Variational Problems

Abstract

Second order ordinary differential equations (SODEs) arise in many areas of natural science. A special class of SODEs come from the variational problems of Lagrange metrics (including Finsler metrics). In general, it is impossible to find explicit solutions to a system of SODEs. Thus, one would like to know the behavior of the solutions based on some data of the system. In the early twentieth century, many geometers had made great effort to study SODEs using geometric methods. Among them are L. Berwald [Bw1, 1926; Bw7, 1947; Bw8, 1947], T.Y. Thomas [Th1, 1925; Th2, 1926; VeTh2, 1926], O. Veblen [Ve1, 1925; Ve2, 1929], J. Douglas [Dg1, 1928; Dg2, 1941], M. S. Knebelman [Kn, 1929], D. Kosambi [Ko2, 1933; Ko3, 1935], E. Cartan [Ca, 1933] and S.S. Chern [Ch2, 1939], etc. In this chapter, we will show that every system of SODEs (also called a semispray) can be studied via the associated system of homogeneous SODEs (also called a spray), and every Lagrange metric can be studied via the associated Finsler metric. Therefore, in the following chapters, we will be mainly concerned with sprays and Finsler metrics, rather than semisprays and Lagrange metrics.