 The first explicit study of the invariance properties of equations in physics is connected with the introduction, in the first half of the nineteenth century, of the transformational approach to the problem of motion in the framework of analytical mechanics. Using the formulation of the dynamical equations of mechanics due to W. R. Hamilton (known as the Hamiltonian or canonical formulation), C. G. Jacobi developed a procedure for arriving at the solution of the equations of motion based on the strategy of applying transformations of the variables that leave the Hamiltonian equations invariant, thereby transforming step by step the original problem into new ones that are simpler but perfectly equivalent (for further details see Lanczos  1949).[5]  Jacobi’s canonical transformation theory, although introduced for the “merely instrumental” purpose of solving dynamical problems, led to a very important line of research: the general study of physical theories in terms of their transformation properties. Examples of this are the studies of invariants under canonical transformations, such as Poisson brackets or Poincaré’s integral invariants; the theory of continuous canonical transformations due to S. Lie; and, finally, the connection between the study of physical invariants and the algebraic and geometric theory of invariants that flourished in the second half of the nineteenth century, and which laid the foundation for the geometrical approach to dynamical problems. The use of the mathematics of group theory to study physical theories was central to the work, early in the twentieth century in Göttingen, of the group whose central figures were F. Klein (who earlier collaborated with Lie) and D. Hilbert, and which included H. Weyl and later E. Noether. We will return later in this section to Weyl (see  Sections 2.1.2,  2.2,  2.5)  and Noether (see  Section 2.1.2).  For more details on these developments see Brading and Castellani (2007).