THE formulation of the laws of mechanics in terms of the Lagrangian, and of Lagrange’s equations derived from it, presupposes that the mechanical state of a system is described by specifying its generalised co-ordinates and velocities. This is not the only possible mode of description, however. A number of advantages, especially in the study of certain general problems of mechanics, attach to a description in terms of the generalised co-ordinates and momenta of the system. The question therefore arises of the form of the equations of motion corresponding to that formulation of mechanics. The passage from one set of independent variables to another can be effected by means of what is called in mathematics Legendre’s transformation. In the present case this transformation is as follows. The total differential of the Lagrangian as a function of co-ordinates and velocities is dL = This expression may be written (40.1) since the derivatives aL/dqi are, by definition, the generalised momenta, and aL/dqi = pi by Lagrange’s equations. Writing the second term in (40.1) as = - Eqi dpi, taking the differential d(piqi) to the left-hand side, and reversing the signs, we obtain from (40.1) The argument of the differential is the energy of the system (cf. §6); expressed in terms of co-ordinates and momenta, it is called the Hamilton’s function or Hamiltonian of the system: (40.2) t The reader may find useful the following table showing certain differences between the nomenclature used in this book and that which is generally used in the English literature. Here Elsewhere Principle of least action Hamilton’s principle Maupertuis’ principle Principle of least action Maupertuis’ principle Action Hamilton’s principal function Abbreviated action Action - -Translators. 131 132 The Canonical Equations §40 From the equation in differentials dH = (40.3) in which the independent variables are the co-ordinates and momenta, we have the equations = (40.4) These are the required equations of motion in the variables P and q, and are called Hamilton’s equations. They form a set of 2s first-order differential equations for the 2s unknown functions Pi(t) and qi(t), replacing the S second- order equations in the Lagrangian treatment. Because of their simplicity and symmetry of form, they are also called canonical equations. The total time derivative of the Hamiltonian is Substitution of qi and pi from equations (40.4) shows that the last two terms cancel, and so dH/dt==Hoo. (40.5) In particular, if the Hamiltonian does not depend explicitly on time, then dH/dt = 0, and we have the law of conservation of energy. As well as the dynamical variables q, q or q, P, the Lagrangian and the Hamiltonian involve various parameters which relate to the properties of the mechanical system itself, or to the external forces on it. Let A be one such parameter. Regarding it as a variable, we have instead of (40.1) dL and (40.3) becomes dH = Hence (40.6) which relates the derivatives of the Lagrangian and the Hamiltonian with respect to the parameter A. The suffixes to the derivatives show the quantities which are to be kept constant in the differentiation. This result can be put in another way. Let the Lagrangian be of the form L = Lo + L’, where L’ is a small correction to the function Lo. Then the corresponding addition H’ in the Hamiltonian H = H + H’ is related to L’ by (H’)p,a - (L’) (40.7) It may be noticed that, in transforming (40.1) into (40.3), we did not include a term in dt to take account of a possible explicit time-dependence