Tullio Levi-Civita

We bring to the reader’s attention a translation of Levi-Civita’s work "Sugli integrali algebrici delle equazioni dinamiche" (1896).
In this work, Levi-Civita exposes some of its results concerning the first integrals of the Lagrangian dynamical systems, which are rational in the velocities. Of a particular historical interest is the fact that here he introduces the concept of Killing tensor and the Killing equation. The Editorial Board is grateful to Professor Sergio Benenti for the translation of the original Italian text and valuable comments.

In this issue we bring to the reader’s attention a translation of Levi-Civita’s work "Sulle trasformazioni delle equazioni dinamiche". This paper, written by Levi-Civita at the onset of his career, is remarkable in many respects. Both the main result and the method developed in the paper brought the author in line with the greatest mathematicians of his day and seriously influenced the further progress of geometry and the theory of integrable systems. Speaking modern language the main result of his paper is the deduction of the general geodesic equivalence equation in invariant form and local classification of geodesically equivalent Riemannian metrics in the case of arbitrary dimension, i.e., metrics having the same geodesics considered as unparameterized curves (this classification problem was formulated by Beltrami in 1865). Levi-Civita’s work produced a great impact on further development of the theory of geodesically equivalent metrics and geodesic mappings, and still remains one of the most important tools in this area of differential geometry.
In this paper the author uses a new method based on the concept of Riemannian connection, which later has been also referred to as the Levi-Civita connection. This paper is truly a pioneering work in the sense that the real power of covariant differentiation techniques in solving a concrete and highly nontrivial problem from the theory of dynamical systems was demonstrated. The author skillfully operates and weaves together many of the most advanced (for that times) algebraic, geometric and analytic methods. Moreover, an attentive reader can also notice several forerunning ideas of the method of moving frames, which was developed a few decades later by E. Cartan.
We hope that the reader will appreciate the style of exposition as well. This work, focused on the essence of the problem and free of manipulation with abstract mathematical terms, is a good example of a classical text of the late 19th century. Owing to this, the paper is easy to read and understand in spite of some different notation and terminology.
The Editorial Board is very grateful to Professor Sergio Benenti for the translation of the original Italian text and valuable comments (see marginal notes at the end of the text, p. 612).