variables at time t, and qt+r, Pt+r their values at another time t +T. The latter are some functions of the former (and involve T as a parameter): If these formulae are regarded as a transformation from the variables Qt, Pt to qt+r, Pttr, then this transformation is canonical. This is evident from the expression ds = for the differential of the action S(qt++, qt) taken along the true path, passing through the points qt and qt++ at given times t and t + T (cf. (43.7)). A comparison of this formula with (45.6) shows that - S is the generating function of the transformation. 46. Liouville’s theorem For the geometrical interpretation of mechanical phenomena, use is often made of phase space. This is a space of 2s dimensions, whose co-ordinate axes correspond to the S generalised co-ordinates and S momenta of the system concerned. Each point in phase space corresponds to a definite state of the system. When the system moves, the point representing it describes a curve called the phase path. The product of differentials dT = dq1 … dqsdp1 dps may be regarded as an element of volume in phase space. Let us now consider the integral I dT taken over some region of phase space, and representing the volume of that region. We shall show that this integral is invariant with respect to canonical transformations; that is, if the variables P, q are replaced by P, Q by a canonical transformation, then the volumes of the corresponding regions of the spaces of P, and P, Q are equal: …dqsdp1…dps = (46.1) The transformation of variables in a multiple integral is effected by the formula I .jdQ1…dQsdP1…dPz = S… I Ddq1 dp1…dps, where (46.2) is the Jacobian of the transformation. The proof of (46.1) therefore amounts to proving that the Jacobian of every canonical transformation is unity: D=1. (46.3) We shall use a well-known property of Jacobians whereby they can be treated somewhat like fractions. “Dividing numerator and denominator” by 0(91, …, qs, P1, Ps), we obtain Another property of Jacobians is that, when the same quantities appear in both the partial differentials, the Jacobian reduces to one in fewer variables,