The Generating Function of a Canonical Transformation

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 − = {Q, P } − . (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- = {Q, P } + , (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 {f, g} ≡ − . (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 = {Q, P } − . (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 {Q, P } = 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. Conversely, a given func- by the coordinate transformation tion F 0(p, Q, t) such that ∂2F 0/∂p∂Q 6= 0, defines a canonical transformation by means of the first two equations in (16). µ ¶ q 2 1 In a similar way, one can consider generating functions de- Q = t − arctan ,P = (p2 + q2). pending on (q, P, t), or (p, P, t) (see, e.g., Refs. 1 to 6). p 2 It should be clear, from the derivation above, that the co- Now {Q, P } = −2(t − arctan q/p), which is also a con- ordinate transformations satisfying condition (9) are not the stant of motion if H = (1/2)(p2 + q2), as above. Further- most general coordinate transformations that leave invariant more, dQ/dt = 0, dP/dt = 0, which can be written in the the form of the Hamilton equations and, by contrast to what form (2) with a new Hamiltonian K = 0. This is not strange, is claimed in some textbooks (e.g., Refs. 3 and 4), the Pois- since in the Hamilton–Jacobi method one finds a transforma- son bracket {Q, P } needs not be a (trivial) constant. (By a tion leading to a new Hamiltonian equal to zero, but this is trivial constant we mean a function whose value is the same usually done with the aid of canonical transformations (the at all points of its domain or, equivalently, a function whose solution of the Hamilton–Jacobi equation is the generating partial derivatives are all identically equal to zero.) A simple function of a canonical transformation to a new set of vari- example is given by the transformation q p ables corresponding to a Hamiltonian equal to zero). For this Q = arctan ,P = p2 + q2. transformation we obtain the relation p µ ¶ One readily finds that the Poisson bracket {Q, P } is equal q £ ¤ P dQ = −2 t − arctan pdq − Hdt − d(pq/2) to (p2 + q2)−1/2, which is not a trivial constant, but is a p constant of the motion if the Hamiltonian is, for instance, H = (1/2)(p2 + q2) (corresponding to a harmonic oscil- [cf. Eqs. (13) and (17)]. lator). Then, the Hamilton equations (1) yield dq/dt = p, The most general coordinate transformation that pre- and dp/dt = −q; therefore, we have, dQ/dt = 1 and serves the form of the Hamilton equations (1) corresponds dP/dt = 0, which can be expressed as the Hamilton equa- to {Q, P } being a constant of the motion. Indeed, making tions (2) if the transformed Hamiltonian is chosen as K = P . use of the definition of the Poisson bracket (7), Eqs. (6), (8), the chain rule, and Eqs. (1) µ ¶ ∂ ∂Q ∂ ∂P ∂P ∂ ∂Q ∂Q ∂ ∂P ∂P ∂ ∂Q ∂Q ∂ ∂H ∂K {Q, P } = + − − = {Q, P } − ∂t ∂q ∂t ∂p ∂p ∂t ∂q ∂p ∂t ∂q ∂q ∂t ∂p ∂q ∂p ∂Q ∂Q µ ¶ µ ¶ µ ¶ ∂P ∂ ∂H ∂K ∂Q ∂ ∂H ∂K ∂P ∂ ∂H ∂K + − {Q, P } + − {Q, P } − − − {Q, P } + ∂p ∂q ∂P ∂P ∂p ∂q ∂Q ∂Q ∂q ∂p ∂P ∂P µ ¶ µ ¶ ∂{Q, P } ∂H ∂Q ∂H ∂P ∂{Q, P } ∂H ∂P ∂H ∂Q = + − + ∂p ∂Q ∂q ∂P ∂q ∂q ∂P ∂p ∂Q ∂p µ ¶ ∂Q ∂ ∂H ∂P ∂ ∂H ∂Q ∂ ∂H ∂P ∂ ∂H + {Q, P } − − + ∂q ∂p ∂Q ∂p ∂q ∂P ∂p ∂q ∂Q ∂q ∂p ∂P ∂Q ∂ ∂K ∂P ∂ ∂K ∂Q ∂ ∂K ∂P ∂ ∂K ∂{Q, P } dp ∂{Q, P } dq − + + − = − − ∂q ∂p ∂Q ∂p ∂q ∂P ∂p ∂q ∂Q ∂q ∂p ∂P ∂p dt ∂q dt µ½ ¾ ½ ¾¶ ½ ¾ ½ ¾ ∂H ∂H ∂K ∂K + {Q, P } Q, + P, − Q, − P, . ∂Q ∂P ∂Q ∂P Now, according to Eq. (11) we have, for instance, ½ ¾ µ ¶ ∂H ∂Q ∂ ∂H ∂Q ∂ ∂H ∂ ∂H Q, = {Q, P } − = {Q, P } ∂Q ∂Q ∂P ∂Q ∂P ∂Q ∂Q ∂P ∂Q and ½ ¾ µ ¶ ∂H ∂P ∂ ∂H ∂P ∂ ∂H ∂ ∂H P, = {Q, P } − = −{Q, P } ; ∂P ∂Q ∂P ∂P ∂P ∂Q ∂P ∂Q ∂P

Rev. Mex. Fis. E 57 (2011) 158–163 THE GENERATING FUNCTION OF A CANONICAL TRANSFORMATION 161 therefore ½ ¾ ½ ¾ By analogy with the case where the number of degrees ∂H ∂H Q, + P, = 0 of freedom is 1, the canonical transformations are deﬁned by ∂Q ∂P the conditions and, similarly, {Qi,Qk} = 0, {P ,P } = 0, {Qi,P } = δi . (23) ½ ¾ ½ ¾ i k k k ∂K ∂K Q, + P, = 0, Then, Eqs. (20) and (22) yield ∂Q ∂P i ∂Q ∂(H − K) ∂Pi ∂(H − K) thus showing that {Q, P } is a constant of motion (cf. Ref. 1). = − , = i (24) ∂t ∂Pi ∂t ∂Q (A shorter proof is given in the appendix.) [cf. Eqs. (12)]. As is well known, Eqs. (23) imply that 2.2. Systems with an arbitrary number of degrees of k k ∂Q ∂Pk ∂Pk ∂Q freedom − = 0, ∂qm ∂qj ∂qm ∂qj When the number of degrees of freedom is greater than 1, the ∂Qk ∂P ∂P ∂Qk k − k = δj , existence of a generating function of any canonical transfor- m m m (25) ∂q ∂pj ∂q ∂pj mation can be demonstrated following essentially the same ∂Qk ∂P ∂P ∂Qk steps as in the preceding subsection. We start assuming that k − k = 0 the set of Hamilton equations ∂pm ∂pj ∂pm ∂pj

i (as a matter of fact, Eqs. (25) are equivalent to Eqs. (23) dq ∂H dpi ∂H = , = − i (18) [1, 4]). Indeed, assuming that Eqs. (23) hold, we have dt ∂pi dt ∂q ∂Qi ∂Qk ∂P (i = 1, 2, . . . , n), is equivalent to the set = {Qi,P } − k {Qi,Qk} ∂qm ∂qm k ∂qm i dQ ∂K dPi ∂K k µ i i ¶ ∂Q ∂Q ∂Pk ∂Q ∂Pk = , = − i , (19) dt ∂Pi dt ∂Q = m j − j ∂q ∂q ∂pj ∂pj ∂q i i µ ¶ where the new coordinates Q and Pi are functions of q , pi, i k i k ∂Pk ∂Q ∂Q ∂Q ∂Q and possibly also of the time. Then, by virtue of the chain − m j − j ∂q ∂q ∂pj ∂pj ∂q rule and Eqs. (18) and (19) we obtain (here and in what fol- µ ¶ lows there is summation over repeated indices) ∂Qi ∂Qk ∂P ∂P ∂Qk = k − k ∂qj ∂qm ∂p ∂qm ∂p ∂K dQi ∂Qi ∂H ∂Qi ∂H ∂Qi j j = − + µ ¶ j j ∂Qi ∂Qk ∂P ∂P ∂Qk ∂Pi dt ∂q ∂pj ∂pj ∂q ∂t − k − k µ ¶ ∂p ∂qm ∂qj ∂qm ∂qj ∂Qi ∂H ∂Qk ∂H ∂P j = + k ∂qj ∂Qk ∂p ∂P ∂p and, in a similar manner, j k j µ ¶ µ ¶ i i k k i k i ∂Q ∂Q ∂Q ∂Pk ∂Pk ∂Q ∂Q ∂H ∂Q ∂H ∂Pk ∂Q = − − + + j ∂p ∂Qk ∂qj ∂P ∂qj ∂t ∂pm ∂q ∂pm ∂pj ∂pm ∂pj j k µ ¶ µ ¶ ∂Qi ∂Qk ∂P ∂P ∂Qk ∂H ∂Qi ∂Qk ∂Qi ∂Qk − k − k , = − j j ∂Qk ∂qj ∂p ∂p ∂qj ∂pj ∂pm ∂q ∂pm ∂q j j µ ¶ µ ¶ k k i i i ∂Pi ∂Pi ∂Pk ∂Q ∂Q ∂Pk ∂H ∂Q ∂Pk ∂Q ∂Pk ∂Q m = m j − m j + j − j + ∂q ∂pj ∂q ∂q ∂q ∂q ∂Pk ∂q ∂pj ∂pj ∂q ∂t µ k k ¶ i ∂Pi ∂Pk ∂Q ∂Q ∂Pk ∂H i k ∂H i ∂Q − − , = {Q ,Q } + {Q ,P } + , (20) j m m k k ∂q ∂q ∂pj ∂q ∂pj ∂Q ∂Pk ∂t µ k k ¶ with the Poisson brackets being now deﬁned by ∂Pi ∂Pi ∂Pk ∂Q ∂Q ∂Pk = j − j ∂pm ∂pj ∂pm ∂q ∂pm ∂q ∂f ∂g ∂f ∂g µ k k ¶ {f, g} ≡ i − i , (21) ∂Pi ∂Pk ∂Q ∂Q ∂Pk ∂q ∂pi ∂pi ∂q − j − , ∂q ∂pm ∂pj ∂pm ∂pj and, similarly, and this set of relations implies Eqs. (25). ∂K ∂H k ∂H ∂Pi Equations (24) and (25) are necessary and sufﬁcient con- − i = k {Pi,Q } + {Pi,Pk} + (22) ∂Q ∂Q ∂Pk ∂t ditions for the local existence of a function F such that i i [cf. Eqs. (6) and (8)]. PidQ − Kdt − pidq + Hdt = dF, (26)

Rev. Mex. Fis. E 57 (2011) 158–163 162 G.F. TORRES DEL CASTILLO as can be readily verified writing the left-hand side of the last Since Eq. (26) does not necessarily hold [see, e.g., equation in terms of the original variables Eq. (17)], in the case of a non-canonical transformation that µ ¶ µ ¶ preserves the form of the Hamilton equations, the integrals ∂Qj ∂Qj ∂Qi P −p dqi+P dp + P +(H − K) dt j i i j i i Zt2 ∂q ∂pi ∂t (p dqi − Hdt) and applying again the standard criterion for the local exact- i ness of a linear differential form. t1 If the 2n variables qi, Qi are functionally independent and t (which is not necessarily the case), Eq. (26) implies that F Z 2 i can be expressed as a function of qi, Qi, and t, in a unique (PidQ − Kdt) way, and t1 ∂F ∂F ∂F do not coincide nor are simply related. However, the actual P = , p = − ,H − K = . (27) i ∂Qi i ∂qi ∂t path followed by the system in phase space corresponds to stationary values of both functionals (this is analogous, for i The independence of the 2n variables q , pi requires instance, to the fact that the point x = 0 is a local minimum that det(∂2F/∂qi∂Qj)6=0. Conversely, given a function for the functions f(x) = x4 and g(x) = 1 − cos x, despite F (qi,Qi, t) satisfying this condition, Eqs. (27) define a lo- the fact that these functions are not the same). cal canonical transformation. i i For the canonical transformations such that the set q , Q Acknowledgements is functionally dependent, one can employ generating functions that depend on other combinations of old and new vari- The author would like to thank the referees for helpful com- i ables; some or all of the q can be replaced by their conju- ments. i gates pi and, similarly, some or all of the Q can be replaced 2n by their conjugates Pi, giving a total of 2 possibilities (not only the four cases considered, e.g., in Ref. 3). Appendix A. Derivation using exterior forms 3. Comparison with other treatments Making use of the properties of the contraction (or interior i The presence of the combinations pidq − Hdt and product) of a vector field with a differential form (see, e.g., i PidQ −Kdt in Eq. (26) is not accidental. It is related to Refs. 7 to 11), one finds that there is only one vector field of the fact that one obtains the Hamilton equations looking for the form i i ∂ ∂ ∂ the path in phase space, q = q (t), pi = pi(t), along which i X = + A i + Bi (A.1) the integral ∂t ∂q ∂pi Zt2 whose contraction with the 2-form i (pidq − Hdt) (28) i ω ≡ dpi ∧ dq − dH ∧ dt (A.2) t1 has a stationary value (usually a minimum) when compared is equal to zero (that is, X ω = 0). In fact, making use of with neighboring paths with the same end points in phase the expressions (A.1) and (A.2), one finds that X ω = 0 is space for t = t1 and t = t2. Since the addition of the dif- equivalent to ferential of any differentiable function F (qi, p , t) to the inte- i ∂H ∂H grand in (28) changes the value of the integral by a term that i A = ,Bi = − i . is the same for all the paths with the same end points in phase ∂pi ∂q space for t = t1 and t = t2, it is right to say that if Hence, the integral curves of X correspond to the solutions

i i of the Hamilton equations (18). PidQ − Kdt = pidq − Hdt + dF, Equations (19) are then equivalent to the condition [which is Eq. (26)] then the Hamilton equations (18) will be X Ω = 0, where equivalent to Eqs. (19). What is wrong to say is that the i Ω ≡ dPi ∧ dQ − dK ∧ dt. (A.3) converse is also true (see, e.g., Refs. 2, 5, and 6), or that i i PidQ − Kdt and pidq − Hdt can only differ by a triv- If we restrict ourselves to canonical transformations, then i i ial constant factor and the differential of a function (see, e.g., Ω = ω, or, equivalently, d(PidQ −Kdt−pidq +Hdt) = 0, Refs. 3 and 4) if Eqs. (18) are equivalent to (19). which implies the local existence of a function F such that Even though Eq. (26) implies that there exists a functional Eq. (26) holds. However, there is an inﬁnite number of relation among F , Qi, qi, and t, another frequent error is to 2-forms Ω of the form (A.3), that do not differ by a triv- conclude that this implies that the 2n variables qi, Qi, are ial multiplicative constant from ω such that, simultaneously, functionally independent (see, e.g., Refs. 4 to 6). X ω = 0 and X Ω = 0 [11].

Rev. Mex. Fis. E 57 (2011) 158–163 THE GENERATING FUNCTION OF A CANONICAL TRANSFORMATION 163

Only in the case of systems with one degree of freedom, any two such 2-forms must be related by Ω = fω, where f is some, nowhere vanishing, real-valued function [11]. Among other things, from Ω = fω it follows that {Q, P } = f [with the Poisson brackets deﬁned by Eq. (7)]. Since ω and Ω are both closed (that is, their exterior derivatives are equal to zero), equation Ω = fω implies that f must obey the condition

df ∧ ω = 0, (A.4) that is µ ¶ µ ¶ µ ¶ ∂f ∂f ∂f ∂H ∂H ∂f ∂H ∂f ∂H ∂f 0 = dq + dp + dt ∧ dp ∧ dq − dq ∧ dt − dp ∧ dt = − + dp ∧ dq ∧ dt. ∂q ∂p ∂t ∂q ∂p ∂q ∂p ∂p ∂q ∂t By virtue of the Hamilton equations (1), this equation holds if and only if f is a constant of the motion, that is Xf = 0 (see the examples at the end of Sec. 2.1).

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Rev. Mex. Fis. E 57 (2011) 158–163