ENSENANZA˜ Revista Mexicana de F´ısica E 57 158–163 DICIEMBRE 2011 The generating function of a canonical transformation G.F. Torres del Castillo Departamento de F´ısica Matematica,´ Instituto de Ciencias, Universidad Autonoma´ de Puebla, 72570 Puebla, Pue., Mexico.´ Recibido el 27 de junio de 2011; aceptado el 3 de octubre de 2011 An elementary proof of the existence of the generating function of a canonical transformation is given. A shorter proof, making use of the formalism of differential forms is also given. Keywords: Canonical transformations; generating function. Se da una prueba elemental de la existencia de una funcion´ generatriz de una transformacion´ canonica.´ Se da tambien´ una prueba mas´ corta, usando el formalismo de formas diferenciales. Descriptores: Transformaciones canonicas;´ funcion´ generatriz. PACS: 45.20.Jj 1. Introduction 2. Canonical transformations In order to present the ideas in a simple way, it is convenient One of the main reasons why the Hamiltonian formalism is to consider firstly the case where there is only one degree of more useful than the Lagrangian formalism is that the set of freedom, which greatly simplifies the derivations. coordinate transformations that leave invariant the form of the Hamilton equations is much wider than the set of coordi- 2.1. Systems with one degree of freedom nate transformations that leave invariant the form of the La- grange equations. Furthermore, each of the so-called canoni- We shall consider a system with one degree of freedom, de- cal transformations leaves invariant the form of the Hamilton scribed by a Hamiltonian function H(q; p; t). This means equations and can be obtained from a single real-valued func- that the time evolution of the phase space coordinates q and tion of 2n + 1 variables, where n is the number of degrees of p is determined by the Hamilton equations freedom of the system, which is therefore called the generat- dq @H dp @H ing function of the transformation. = ; = ¡ : (1) dt @p dt @q The proof of the existence of a generating function for We want to find the coordinate transformations, an arbitrary canonical transformation given in most standard Q=Q(q; p; t), P =P (q; p; t), that maintain the form of the textbooks is usually based on the calculus of variations (see, Hamilton equations (1). That is, we want that Eqs. (1) be e.g., Refs. 1 to 6), which allows one to obtain the basic rela- equivalent to tions quickly. The aim of this paper is to give a straightfor- dQ @K dP @K ward, elementary derivation of the existence of the generating = ; = ¡ ; (2) function of a canonical transformation, not based on the cal- dt @P dt @Q culus of variations. One of the advantages of the proof given where K may be the original Hamiltonian H expressed in here is that it allows one to see clearly the assumptions in- terms of the new coordinates or another function. (The last volved, by contrast with the more diffuse proof usually given possibility is relevant since it turns out that the new Hamilto- in the textbooks, and to realize that the canonical transfor- nian can be made equal to zero by means of a suitable trans- mations are not the most general transformations that leave formation.) invariant the form of the Hamilton equations. In Sec. 2, the Assuming that the transformation Q = Q(q; p; t), definition of a canonical transformation is briefly reviewed in P = P (q; p; t) is differentiable and can be inverted (that is, it order to derive the basic equations that lead to the existence is possible to find q and p in terms of Q, P , and t and, there- of the generating function of the transformation. In Sec. 3 we fore, H can be viewed also as a function of Q, P , and t), mak- point out some of the frequent errors contained in the proofs ing use repeatedly of the chain rule and of Eqs. (1) and (2) we given in some of the standard textbooks. For those readers ac- find that quainted with the formalism of (exterior) differential forms, @K dQ @Q @H @Q @H @Q = = ¡ + (3) a considerably shorter proof is given in the appendix. The @P dt @q @p @p @q @t simplicity of this latter proof may serve as an invitation to µ ¶ @Q @H @Q @H @P learn the language of differential forms for those not already = + (4) familiar with it. @q @Q @p @P @p THE GENERATING FUNCTION OF A CANONICAL TRANSFORMATION 159 for any pair of functions f; g, if and only if Eq. (9) holds. In µ ¶ fact, making use of the chain rule, one can readily show that @Q @H @Q @H @P @Q ¡ + + (5) µ ¶ @p @Q @q @P @q @t @f @g @f @g @f @g @f @g ¡ = fQ; P g ¡ : (11) µ ¶ @q @p @p @q @Q @P @P @Q @H @Q @P @Q @P @Q = ¡ + @P @q @p @p @q @t Thus, restricting ourselves to coordinate transformations @H @Q satisfying Eq. (9), but allowing them to involve the time ex- = fQ; P g + ; (6) plicitly, Eqs. (6) and (8) yield @P @t @Q @(H ¡ K) @P @(H ¡ K) where we have made use of the definition of the Poisson = ¡ ; = : (12) brackets @t @P @t @Q @f @g @f @g ff; gg ´ ¡ : (7) Now, it turns out that Eqs. (9) and (12) are necessary and suf- @q @p @p @q ficient conditions for the local existence of a function F such In a similar way, we obtain that P dQ ¡ Kdt ¡ pdq + Hdt = dF; (13) @K @H @P = fQ; P g ¡ : (8) as can be seen writing the left-hand side of the last equation @Q @Q @t as µ ¶ µ ¶ Now we have two possibilities: either the Hamilto- @Q @Q @Q P ¡ p dq + P dp + P + (H ¡ K) dt nian K is essentially the original Hamiltonian¡ H, ex- @q @p @t pressed in terms¢ of the new variables [that is, K Q(q; p; t), P (q; p; t); t =H(q; p; t)], or K differs from H. In the first and applying the standard criterion for a linear (or Pfaffian) case, Eqs. (6) and (8) will hold, independent of the Hamilto- differential form to be exact. For instance, by considering the nian H, if and only if coefficients of dq and dt (recalling that q, p, and t are treated as three independent variables), we have fQ; P g = 1 (9) µ ¶ µ ¶ @ @Q @ @Q P + (H ¡ K) ¡ P ¡ p and the coordinate transformation does not involve the time, @q @t @t @q Q = Q(q; p), P = P (q; p). Then, Eq. (9) is the necessary @P @Q @P @Q @(H ¡ K) = ¡ + and sufficient condition for the local existence of a function @q @t @t @q @q F such that @P @(H ¡ K) @Q @(H ¡ K) @(H ¡ K) P dQ ¡ pdq = dF: (10) =¡ ¡ + =0: @q @P @q @Q @q (That is, the function F may not be defined in all the phase If q and Q are functionally independent, then the function space, we can only ensure its existence in some neighbor- F appearing in Eq. (13) can be expressed in terms of q, Q, hood of each point of the phase space.) In fact, writing the and t (in a unique way), and from Eq. (13) it follows that left-hand side of Eq. (10) in the equivalent form µ ¶ @F @F @F @Q @Q P = ; p = ¡ ;H ¡ K = ; (14) P ¡ p dq + P dp; @Q @q @t @q @p and, necessarily, @2F=@q@Q 6= 0 (otherwise q and p would one finds that the condition not be independent). Conversely, given a function F (q; Q; t) µ ¶ µ ¶ 2 @ @Q @ @Q such that @ F=@q@Q 6= 0, Eqs. (14) can be locally inverted P = P ¡ p to find Q and P in terms of q, p, and t. In this way, F is a @q @p @p @q generating function of a canonical transformation. is equivalent to Eq. (9) [1]. If q and Q are functionally dependent (that is, Q can be Even though more general transformations are also possi- expressed as a function of q and t only, or q can be expressed ble (see below), attention is restricted to the transformations as a function of Q and t only), the function F appearing in satisfying Eq. (9), also when the coordinate transformation Eq. (13) can be written in infinitely many ways in terms of q, involves the time explicitly. The coordinate transformations Q, and t, and the first two equations in (14) make no sense satisfying Eq. (9) are called canonical transformations. One (since, e.g., keeping q and t constant in the partial differentia- good reason to consider only canonical transformations is tion with respect to Q, would make Q also constant). In such that the Poisson brackets (7) are invariant under these trans- a case, the variables p and Q (as well as P and q) are neces- formations, in the sense that sarily functionally independent (otherwise q and p would be dependent). Then, writing Eq. (14) in the equivalent form @f @g @f @g @f @g @f @g ¡ = ¡ ; 0 @q @p @p @q @Q @P @P @Q P dQ ¡ Kdt + qdp + Hdt = dF ; (15) Rev. Mex. Fis. E 57 (2011) 158–163 160 G.F. TORRES DEL CASTILLO where F 0 ´ F + pq, it follows that the generating function In place of an equation of the form (13), in this case one finds F 0 can be expressed in a unique way as a function of Q, p, the relation and t, and the canonical transformation is determined by 2 2 ¡1=2£ ¤ @F 0 @F 0 @F 0 P dQ¡Kdt = 2(p +q ) pdq¡Hdt¡d(pq=2) : (17) P = ; q = ;H ¡ K = (16) @Q @p @t A second example, related to the previous one, is given and, necessarily, @2F 0=@p@Q 6= 0.