A number of applications of S. Eilenberg and S. Maclane's cohomology theory of groups to the kinematical groups of physics are presented. Within this field, we apply the theory of group exten sions by Abelian and non-Abelian kernels to the study of the algebraic structures of the Galilei, Static and Carroll groups, and introduce to physics the mathematical concepts of group enlargements and prolongations. The global algebraic structures of the kinematical groups are analysed in depth using these tools and a generalisation of kinematical groups is attempted. The use of the methods of homological algebra in classical mechanics is discussed from the new view point of Lagrangian mechanics introduced by Lévy-Leblond. In this direction two advances are made. Homological algebra is introduced to the study of Hamilton's principle and then a reformulation of Levy-Leblond's free Lagrangian mechanics is obtained. Whilst the above author concentrates on a certain second cohomology group, we see that it is a first cohomology group which is more relevant to this approach. The group theoretic discussion of non-inertial motions is initiated using the theory of the loop prolongations of a group Q by a group K, where a loop is a 'non-associative group'. Our preliminary results enable us to give a cohomological description of constant Newtonian acceleration
