Projective transformations and symmetries of differential equation

The group properties of the equations of geodesics on a pseudo-Riemannian manifold $M^n$ are considered, in particular, when these are written as a system of second-order differential equations (resolved with respect to the second derivatives) with third-degree polynomials in the derivatives of the unknown function on the right-hand sides. Each point symmetry of such systems is proved to be a projective transformation. A connection between projective transformation in $M^n$ and symmetries of Hamiltonian systems and Lie–Bäcklund transformations of Hamilton–Jacobi equation with quadratic Hamiltonians is discovered. This provides tools for developing a systematic geometric approach to defining and investigating point and non-point symmetries of large classes of differential equations and partial differential equations and to obtaining their solutions. The dimension of the maximal symmetry group for system of second-order ordinary differential equations resolved with respect to the higher derivatives is found, and this group is proved to be the projective group. As a consequence, the dimension of the maximal symmetry group of the Newton equations is found. In case of three spatial dimensions this group (which is a 24-dimensional projective group) is proved to have as a subgroup the Poincaré group, which is fundamental for special relativity theory.