Dynamical symmetries and conservation laws

Finding all dynamical symmetries of a given system of differential equations essentially means finding all of its first integrals and thus finding the general solution; this is one of the results of the analysis carried through in Section 12.3. So there is no hope of finding all dynamical symmetries in the general case, but with some luck we may find some of them if we make an appropriate ansatz for the components of the generator X (in fact, the situation is the complete analogue of the search for an integrating factor – or a point symmetry – for a first order differential equation). This has been done for several systems of second order differential equations that are derivable from a Lagrangian (compare also Section 10.3), and we shall concentrate on those systems from now on.

From Section 10.3 we know that if a Lagrangian exists, we could assign a first integral to every point symmetry that was also a Noether symmetry and that this first integral could be determined rather easily if the generator of the Noether symmetry was known. To make use of a dynamical symmetry, it would be very helpful if a similar procedure could be applied here.

Indeed, it is possible to define a subclass of the dynamical symmetries, the so-called Cartan symmetries, and to show that there is a one-to-one correspondence between gauged (ξ, = 0) Cartan symmetries and first integrals ϕ and that one can construct the first integral once the Cartan symmetry is known.