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p P Q q t g 0
G G
t t = t 0 Fig. 10.2 We now prove the second part of the theorem. Since the difference ω − Ω is exact we have γ 0
i =1 p i dq i − Γ 0 l i =1 P i dQ i = Γ 0 (ω − Ω ) = 0,
and the transformation is canonical. Therefore the characteristic directions of the form ω coincide, after the transformation, with those of the form Ω =
i =1 P i dQ i − K dt, where K is the new Hamiltonian. On the other hand, the characteristic directions of ω coincide with those of Ω + d
F, and hence of Ω . In addition Ω − Ω = (K
− K) dt and the coincidence of characteristics implies that K − K may depend only on t. Hence, following our convention, K = K. Example 10.19 We consider again Example 10.11 in the light of the results of this section. By equation (10.49), the Lie condition (10.84) can be written as p ˜
dq − P ˜dQ = (P − at)(dQ + t dP ) − P dQ = ˜df(P, Q, t), from which it follows that f (P, Q, t) = t P 2
− at 2 P − atQ + f 1 (t), where f 1 is an arbitrary function of time. The condition (10.88) for the transformation (10.49), taking into account (10.50), can be written as (P − at)(dQ + P dt + t dP − at dt) − P dQ + − P 2 2 − aQ dt = dF(P, Q, t), 10.3 Analytical mechanics: canonical formalism 361 and after some simple manipulations we find F(P, Q, t) = 1 2 tP 2 − at 2 P − atQ + 1 3 a 2 t 3 . We conclude this section by proving that the Hamiltonian flow defines a canonical transformation. Let H(p, q, t) be a Hamiltonian function, and consider the associated Hamiltonian flow x = S t X: p i = p i (P, Q, t), q i
i (P, Q, t), (10.89) where i = 1, . . . , l. Equations (10.89) are therefore the solutions of the system of equations ∂p i ∂t = − ∂H ∂q i , ∂q i ∂t = ∂H ∂p i , (10.90) with initial conditions p i (0) = P
i , q
i (0) = Q
i , i = 1, . . . , l. By the theorem of existence, uniqueness and continuous dependence on the initial data for ordinary differential equations (see Appendix 1) equation (10.89) defines a coordinate transformation which is regular and invertible. T heorem 10.13 The Hamiltonian flow (10.89) is a time-dependent canonical transformation, that at every time instant t maps X to S t X. In addition, the new Hamiltonian associated with H in the variables X is K ≡ 0.
Proof We verify that the Lie condition (10.84) is satisfied, with f (P, Q, t) = t 0 l j =1 p j (P, Q, τ ) ∂q j ∂t (P, Q, τ ) − H(p(P, Q, τ), q(P, Q, τ), τ) dτ. (10.91) By Remark 10.18, it is enough to show that for every i = 1, . . . , l we have ∂f ∂P i (P, Q, t) = l j =1 p j (P, Q, t) ∂q j ∂P i (P, Q, t), ∂f ∂Q i (P, Q, t) = l j =1 p j (P, Q, t) ∂q j ∂Q i (P, Q, t) − P i . We prove the second relation. The first one can be shown in an analogous manner. We have
∂f ∂Q i = t 0 l j =1 ∂p j ∂Q i ∂q j ∂t + p
j ∂ 2 q j ∂t∂Q i − ∂H ∂p j ∂p j ∂Q i − ∂H ∂q j ∂q j ∂Q i dτ, 362 Analytical mechanics: canonical formalism 10.3 but since (10.89) is the transformation generated by the Hamiltonian flow, it follows from equations (10.90) that ∂f ∂Q i = t 0 l j =1 ∂p j ∂Q i ∂q j ∂t + p j ∂ 2 q j ∂t∂Q i − ∂q j ∂t ∂p j ∂Q i + ∂p j ∂t ∂q j ∂Q i dτ = t 0 ∂ ∂t l j =1 p j ∂q j ∂Q i dτ = l j =1 p j (P, Q, t) ∂q j ∂Q i (P, Q, t) − l j =1 p j (P, Q, 0) ∂q j ∂Q i (P, Q, 0) = l j =1 p j (P, Q, t) ∂q j ∂Q i (P, Q, t) − l j =1 P j δ ji = l j =1 p j (P, Q, t) ∂q j ∂Q i (P, Q, t) − P i
By what we have just computed, ˜ df = l i =1 p i ˜ dq i − P i dQ i , while from (10.83) it obviously follows that df = ˜ df +
∂f ∂t dt = l i =1 p i ˜ dq i − P i ˜ dQ i + l j =1 p j ∂q j ∂t − H dt
= l i =1 (p i dq i − P i dQ i ) − H dt.
Taking into account Theorem 10.6, it follows from this that the new Hamiltonian associated with H is exactly K ≡ 0. Remark 10.19 From the expression (10.91) for f , since ˙ p i = ∂p i /∂t and ˙ q i = ∂q i /∂t, we see that f (P, Q, t) is the Hamiltonian action A(P, Q, t) (see (9.43)) computed by an integration along the Hamiltonian flow (10.89), i.e. the natural motion. Recalling the result of Corollary 10.1, we can now state that the canonical transformations depending on time are all and exclusively the Hamiltonian flows. If we apply the canonical transformation x = x(x ∗ , t) generated by the Hamiltonian H(x, t), to a system with Hamiltonian H ∗ (x ∗ , t), we obtain the new Hamiltonian K ∗
H ∗ (x, t) + H(x, t) (here H plays the role of the function indicated by K 0 in the previous section). Consider now the Hamiltonian flow x = S t X, with
Hamiltonian H(x, t). The inverse transformation, mapping S t X in X for every t, corresponds to the retrograde motion (with Hamiltonian −H) and it is naturally 10.3 Analytical mechanics: canonical formalism 363 canonical. For the canonical transformation x = S t X the variables X play the role of constant canonical coordinates ( ˙ X = 0). In agreement with this fact, we note that the composition of the two flows yields the Hamiltonian K(X, t) = 0 and therefore precisely constant canonical coordinates. As an example, note that the transformation (10.49) is the flow with Hamiltonian H = p 2 /2 + aq. This is independent of time, and hence it is a constant of the motion, implying that p 2 /2+aq = P 2 /2+aQ. This is the equation for the trajectories, travelled ‘forwards’ (P, Q) → (p, q) through the flow with Hamiltonian H(p, q), and ‘backwards’ (p, q) → (P, Q) with Hamiltonian (10.50), i.e. −H(P, Q). The superposition of the two yields (P, Q) → (P, Q) for every t, and hence ˙P = ˙Q = 0 (corresponding to the null Hamiltonian). Remark 10.20 The apparent lack of symmetry between the condition l i =1 (p i dq i − P i dQ i ) = d F, where F is independent of t, for a transformation to be completely canonical, and the relation l i
(p i dq i − P
i dQ i ) + (K − H) dt = dF, where F depends also on t, for a time-dependent transformation to be canonical, can be eliminated by using a significant extension of the Hamiltonian formalism. Indeed, given a non-autonomous Hamiltonian system H(p, q, t), we consider, in addition to the canonical equations (10.90), the equations (see (8.26)) − ˙
H = − dH dt = − ∂H ∂t , ˙t = 1. (10.92)
The system of equations (10.90), (10.92) corresponds to the canonical equations for the Hamiltonian H : R 2l+2
→ R, H(p, π, q, τ) = H(p, q, τ) + π, (10.93) where
π = −H, τ = t, (10.94) and hence the Hamiltonian and time are considered as a new pair of canonically conjugate variables. This is possible since ∇ p H = ∇ p H, ∇ q H = ∇ q H and
˙π = − ∂ H ∂τ = − ∂H ∂t , ˙τ =
∂ H ∂π = 1. 364 Analytical mechanics: canonical formalism 10.4 By (10.94) we also have that H = 0, and the Poincar´e–Cartan form (10.77) becomes
l i =1 p i dq i − H dt =
l i =1 p i dq i + π dτ =
l +1 i =1 p i dq i , (10.95) where we set p l +1
l +1 = τ . The canonical transformations (10.81) are therefore always completely canon- ical in R 2l+2 , and they associate with the variables (p, π, q, τ ) new variables (P, Π , Q, T ), with the constraint T = τ . The Hamiltonian H is always zero. Conversely, transformations such as τ = a(T ), π =
1 a (T )
Π (10.96)
can be included in the canonical formalism, since π dτ =
1 a (T )
Π a (T ) dT = Π dT.
The effect of equation (10.96) is a re-parametrisation of time, and by using the fact that it is canonical one can show that the canonical structure of Hamilton’s equations is preserved, by appropriately rescaling the Hamiltonian H = −π.
10.4 Generating functions In the previous sections we completely described the class of canonical trans- formations. We now study a procedure to generate all canonical transformations. As we saw in the previous section, the Lie condition (10.84), or its equival- ent formulation (10.88), is a necessary and sufficient condition for a coordinate transformation to be canonical. In the form (10.88), it allows the introduction of an efficient way to construct other canonical transformations. Assume that p = p(P, Q, t), q = q(P, Q, t) (10.97)
defines a canonical transformation in an open domain of R 2l , with inverse P = P(p, q, t), Q = Q(p, q, t). (10.98) A canonical transformation of the type (10.97) satisfying det ∂q i ∂P j = / 0 (10.99)
is called free. Applying the implicit function theorem to the second of equations (10.97), the condition (10.99) ensures that the variables P can be naturally expressed as functions of the variables q, Q, as well as of time. Therefore, if P = ˆ
P(q, Q, t), (10.100)
10.4 Analytical mechanics: canonical formalism 365 by substituting this relation into the first of equations (10.97) we find p = ˆ p(q, Q, t). (10.101) The condition (10.88) l i
p i dq i − H dt −
l i =1 P i dQ i − K dt = dF can therefore be written l i =1 ˆ p i (q, Q, t) dq i − H(q, ˆp(q, Q, t), t) dt − l i =1 ˆ P i (q, Q, t) dQ i − K( ˆP(q, Q, t), Q, t) dt = dF (q, Q, t), (10.102) where the variables (q, Q) are considered to be independent and F (q, Q, t) is obtained from F(P, Q, t) through equation (10.100). From (10.102) it follows that p i
∂F ∂q i , (10.103)
P i = − ∂F ∂Q i , (10.104) K = H + ∂F ∂t , (10.105)
where i = 1, . . . , l. Equation (10.104) shows that the matrix − (∂q i
j ) is the inverse matrix of ∂ 2
i ∂Q j ) . Therefore the condition (10.99) is clearly equivalent to requiring that
det ∂ 2 F ∂q i ∂Q j = / 0. (10.106)
We now follow the converse path, starting from the choice of a function of the type (10.106). D efinition 10.10 A function F (q, Q, t) satisfying condition (10.106) is called a generating function (of the first kind, and it is often denoted by F = F 1 ) of the canonical transformation defined implicitly by equations (10.103)–(10.105). Remark 10.21 Given the generating function F , equations (10.103)–(10.105) define the canon- ical transformation implicitly. However the condition (10.106) ensures that the variables Q can be expressed as functions of (q, p) and of time t, by invert- ing equation (10.103). The expression of P as a function of (q, p) and of the time t can be obtained by substituting the relation Q i = Q i (q, p, t) into equation (10.104). The invertibility of the transformation thus obtained is again guaranteed
366 Analytical mechanics: canonical formalism 10.4 by the implicit function theorem. Indeed, equation (10.106) also ensures that it is possible to express q = q(Q, P, t) by inverting (10.104). Substituting these into equation (10.103) we finally find p = p(Q, P, t). Example 10.20 The function F (q, Q) = mω/2q 2 cot Q generates a canonical transformation p = √ 2P ωm cos Q, q = 2P ωm sin Q, which transforms the Hamiltonian of the harmonic oscillator H(p, q) = p 2 2m + mω 2 q 2 2 into
K(P, Q) = ωP. Example 10.21 The identity transformation p = P , q = Q is not free. Hence it does not admit a generating function of the first kind. After setting x = (p, q) and X = (P, Q), we see that a generating function can also depend on x m 1
m l , X n 1 , . . . , X n l for an arbitrary choice of the indices m i and n i (all different). We quickly analyse all possible cases. D efinition 10.11 A function F (q, P, t) satisfying the condition det ∂ 2 F ∂q i ∂P j = / 0 (10.107)
is called a generating function of the second kind (and it is often denoted by F = F
2 ) of the canonical transformation implicitly defined by p i
∂F ∂q i , i = 1, . . . , l, (10.108) Q i = ∂F ∂P i , i = 1, . . . , l. (10.109) Example 10.22 Point transformations (see Example 10.9) Q = Q(q, t) 10.4 Analytical mechanics: canonical formalism 367 are generated by F 2 (q, P, t) = l i =1 P i Q i (q, t).
Setting Q = q we find that F 2 = l i =1 P i q i is the generating function of the identity transformation. D efinition 10.12 A function F (p, Q, t) which satisfies the condition det ∂ 2 F ∂p i ∂Q j = / 0 (10.110)
is called a generating function of the third kind (and it is often denoted by F = F
3 ) of the canonical transformation implicitly defined by q i
− ∂F ∂p i , i = 1, . . . , l, (10.111) P i = − ∂F ∂Q i , i = 1, . . . , l. (10.112) Example 10.23 It is immediate to check that the function F (p, Q) = −p(e
Q − 1) generates the canonical transformation P = p(1 + q), Q = log(1 + q). D efinition 10.13 A function F (p, P, t) which satisfies the condition det ∂ 2 F ∂p i ∂P j = / 0 (10.113)
is called a generating function of the fourth kind (and it is often denoted by F = F
4 ) of the canonical transformation implicitly defined by q i
− ∂F ∂p i , i = 1, . . . , l, (10.114) Q i = ∂F ∂P i , i = 1, . . . , l. (10.115) Example 10.24 The canonical transformation of Example 10.8, exchanging the coordinates and the kinetic momenta, admits as generating function F (p, P) = l i
p i P i .
368 Analytical mechanics: canonical formalism 10.4 T
1 , F
2 , F
3 and F
4 satisfy, respectively, l i
(p i dq i − P
i dQ i ) + (K − H) dt = dF 1 (q, Q, t), (10.116) l i =1 (p i dq i + Q i dP i ) + (K − H) dt = dF 2 (q, P, t), (10.117) l i =1 ( −q i dp i − P i dQ i ) + (K
− H) dt = dF 3 (p, Q, t), (10.118) l i =1 ( −q i dp i + Q i dP i ) + (K
− H) dt = dF 4 (p, P, t). (10.119) If a canonical transformation admits more than one generating function of the previous kinds, then these are related by a Legendre transformation: F 2 = F 1 + l i =1 P i Q i , F 3 = F
1 − l i =1 p i q i , F 4 = F 1 − l i =1 p i q i + l i =1 P i Q i = F 2 − l i =1 p i q i = F
3 + l i =1 P i Q i . (10.120)
Proof The first part of the theorem is a consequence of Definitions 10.10–10.13. The proof of the second part is immediate, and can be obtained by adding or subtracting l i
P i Q i and
l i =1 p i q i from (10.116). Remark 10.22 At this point it should be clear how, in principle, there exist 2( 2l l
kinds of generating functions, each corresponding to a different arbitrary choice of l variables among q, p and of l variables among Q, P. However, it is always possible to reduce it to one of the four previous kinds, by taking into account that the exchanges of Lagrangian coordinates and kinetic momenta are canonical transformations (see Example 10.8). The transformations associated with generating functions exhaust all canonical transformations. T heorem 10.15 It is possible to associate with every canonical transformation a generating function, and the transformation is completely canonical if and only if its generating function is time-independent. The generating function is of one of the four kinds listed above, up to possible exchanges of Lagrangian coordinates with kinetic moments. 10.4 Analytical mechanics: canonical formalism 369 Proof
Consider a canonical transformation, and let F the function associated with it by Theorem 10.12. If it is possible to express the variables p, P as functions of q, Q, and hence if (10.99) holds, then, as we saw at the beginnning of this section, it is enough to set F 1 (q, Q, t) = F( ˆP(q, Q, t), Q, t) and the conditions of Definition 10.10 are satisfied. If, on the other hand, we have det ∂q
∂Q j = / 0, (10.121)
we can deduce Q = ˆ Q(q, P, t) from the second of equations (10.97) and, by substitution into the first of equations (10.97), we find that the variables p can also be expressed through q, P. Hence we set F 2
F(P, ˆ Q(q, P, t), t) + l i
P i ˆ Q i (q, P, t). The condition (10.107) is automatically satisfied, since ∂ 2 F /∂q i ∂P j is the
inverse matrix of (∂q i /∂Q j ). Analogously, if det ∂p i ∂P j = / 0, (10.122)
the variables q, P can be expressed through p, Q, and we set F 3 (p, Q, t) = F( ˆP(p, Q, t), Q, t) − l i
p i ˆ q i (p, Q, t). Then the conditions of Definition 10.12 are satisfied. Finally, if det ∂p
∂Q j = / 0, (10.123)
by expressing q, Q as functions of p, P, we find that the generating function is given by
F 4 (p, P, t) = F(P, ˆ Q(p, P, t), t) − l
=1 p i ˆ q i (p, P, t) + l i =1 P i ˆ Q i (p, P, t). It is always possible to choose l variables among p, q and l variables among P, Q as independent variables. As a matter of fact, the condition that the Jacobian
370 Analytical mechanics: canonical formalism 10.4 matrix of the transformation is symplectic, and therefore non-singular, guarantees the existence an l ×l submatrix with a non-vanishing determinant. If the selected independent variables are not in any of the four groups already considered, we can proceed in a similar way, and obtain a generating function of a different kind. On the other hand, it is always possible to reduce to one of the previous cases by a suitable exchange of variables. Remark 10.23 An alternative proof of the previous theorem, that is maybe more direct and certainly more practical in terms of applications, can be obtained simply by remarking how conditions (10.99), (10.121)–(10.123) ensure that the Lie condi- tion can be rewritten in the form (10.116)–(10.119), respectively. The functions F 1 , . . . , F 4 can be determined by integration along an arbitrary path in the domain of definition and the invertibility of the transformation. Example 10.25 Consider the canonical transformation p = 2e
t P Q log P, q = e −t
defined in D = {(P, Q) ∈ R 2 |P > 0, Q ≥ 0} ⊂ R 2 . Evidently it is possible to choose (q, P ) as independent variables and write p = 2e
2t q log P,
Q = e 2t q 2 P . The generating function F 2 (q, P, t) can be found, for example, by integrating the differential form ˆ p(q, P, t) dq + ˆ Q(q, P, t) dP along the path γ = {(x, 1)|0 ≤ x ≤ q} ∪ {(q, y)|1 ≤ y ≤ P } in the plane (q, P ). Since along the first horizontal part of the path γ one has p(x, 1, t) ≡ 0 (this simplification motivates the choice of the integration path γ), we have F 2
2t q 2 P 1 dy y + ˜
F 2 (t) = e 2t q 2 log P + ˜ F 2 (t), where ˜
F 2 is an arbitrary function of time. Remark 10.24 Every generating function F is defined up to an arbitrary additive term, a function only of time. This term does not change the transformation gener- ated by F , but it modifies the Hamiltonian (because of (10.105)) and it arises from the corresponding indetermination of the difference between the Poincar´ e– Cartan forms associated with the transformation (see Remark 10.18). Similarly 10.5 Analytical mechanics: canonical formalism 371 to what has already been seen, this undesired indetermination can be overcome by requiring that the function F does not contain terms that are only functions of t.
We conclude this section by proving a uniqueness result for the generating function (once the arbitrariness discussed in the previous remark is resolved). P roposition 10.4 All the generating functions of a given canonical transform- ation, depending on the same group of independent variables, differ only by a constant. Proof Consider as an example the case of two generating functions F (q, Q, t) and G(q, Q, t). The difference F − G satisfies the conditions ∂ ∂q
(F − G) = 0, ∂ ∂Q
(F − G) = 0, for every i = 1, . . . , l. Hence, since by Remark 10.24 we have neglected additive terms depending only on time, F − G is necessarily constant. 10.5
Poisson brackets Consider two funtions f (x, t) and g(x, t) defined in R 2l × R with sufficient regularity, and recall the definition (10.16) of a standard symplectic product. D efinition 10.14 The Poisson bracket of the two functions, denoted by {f, g}, is the function defined by the symplectic product of the gradients of the two functions: {f, g} = (∇ x f )
T I∇ x g. (10.124)
Remark 10.25 If x = (p, q), the Poisson bracket of two functions f and g is given by {f, g} = l i =1 ∂f ∂q i ∂g ∂p i − ∂g ∂q i ∂f ∂p i . (10.125) Remark 10.26 Using the Poisson brackets, Hamilton’s equations in the variables (p, q) can be written in a perfectly symmetric form as ˙ p
= {p i , H }, ˙ q i = {q i , H }, i = 1, . . . , l. (10.126)
Remark 10.27 From equation (10.125) we derive the fundamental Poisson brackets {p i
j } = {q
i , q
j } = 0, {q i , p
j } = −{p
i , q
j } = δ
ij . (10.127) 372 Analytical mechanics: canonical formalism 10.5 Example 10.26 If we consider the phase space R 6 of a free point particle, if L 1 , L
2 and L
3 are
the three components of its angular momentum, and p 1 , p 2 , p
3 are the kinetic momenta, conjugate with the Cartesian cordinates of the point, we have: {p 1 , L 3 } = −p 2 , {p 2 , L
3 } = p
1 , {p 3 , L
3 } = 0,
and similarly for L 1 and L 2 . Using the Ricci tensor ijk , the previous relations take the more concise form {p i , L j } = ijk p k ( ijk
= 0 if the indices are not all different, otherwise ijk
= ( −1)
n , where n is the number of permutations of pairs of elements to be performed on the sequence {1, 2, 3} to obtain {i, j, k}). It can be verified in an analogous way that {L i
j } =
ijk L k , and that
{L i , L 2 } = 0,
where L 2 = L 2 1 + L 2 2 + L 2 3 . The Poisson brackets are an important tool, within the Hamiltonian formalism, for the analysis of the first integrals of the motion (also, as we shall see, to characterise the canonical transformations). Indeed, let H : R 2l × R → R, H = H(x, t) be a Hamiltonian function and consider the corresponding canonical equations ˙x = I∇
H, (10.128)
with initial conditions x(0) = x 0 . Suppose that the solution of Hamilton’s equations can be continued for all times t ∈ R, for any initial condition. In this case, the Hamiltonian flow x(t) = S t (x 0 ) defines an evolution operator U t acting on the observables of the system, i.e. on every function f : R 2l × R → R,
f = f (x, t): (U t f )(x 0 , 0) = f (S t x 0 , t) = f (x(t), t). (10.129)
D efinition 10.15 A function f(x, t) is a first integral for the Hamiltonian flow S t
0 ∈ R
2l and t
∈ R, it holds that f (S
t x 0 , t) = f (x 0 , 0). (10.130) 10.5 Analytical mechanics: canonical formalism 373 The total derivative of f with respect to time t, computed along the Hamiltonian flow S t , is given by df dt = ∂f ∂t + ( ∇ x f ) T ˙x =
∂f ∂t + ( ∇ x f ) T I∇ x H. Then using equation (10.124) we have df dt
∂f ∂t + {f, H}, (10.131)
which yields the following. T heorem 10.16 A function f(x), independent of time, is a first integral for the Hamiltonian flow S t if and only if its Poisson bracket with the Hamiltonian vanishes. This characterisation of first integrals is one of the most important properties of the Poisson brackets. However, since Definition 10.14 is made with reference to a specific coordinate system, while a first integral depends only on the Hamilto- nian flow and is evidently invariant under canonical transformations, we must consider the question of the invariance of the Poisson brackets under canonical transformations. T heorem 10.17 The following statements are equivalent. (1) The transformation x = x(X, t), (10.132) is canonical. (2) For every pair of functions f (x, t) and g(x, t), if F (X, t) = f (x(X, t), t) and G(X, t) = g(x(X, t), t) are the corresponding transforms, then {f, g} x
{F, G} X (10.133) at every instant t. Here {f, g}
x indicates the Poisson bracket computed with respect to the original canonical variables x = (p, q), and {F, G}
X indicates that computed with respect to the new variables X = (P, Q). (3) For every i, j = 1, . . . , l and at every instant t it holds that {P i
j } x = {Q i , Q j } x = 0,
{Q i , P j } x = δ ij , (10.134) i.e. the transformation (10.132) preserves the fundamental Poisson brackets. Proof We start by checking that (1) ⇒ (2). We know that a transformation is canonical if and only if its Jacobian matrix J = ∇
X 374 Analytical mechanics: canonical formalism 10.6 is at every instant a symplectic matrix. Using equation (10.124) and recalling the transformation rule for the gradient, we find ∇ x f = J T ∇ X F ,
{f, g} x = ( ∇ x f ) T I∇ x g = (J T ∇ X F )
T IJ T ∇ X G = ( ∇ X F ) T J IJ T ∇ X G = {F, G}
X . That (2) ⇒ (3) is obvious ((3) is a special case of (2)). To conclude, we prove then that (3) ⇒ (1). For this it is enough to note that equations (10.134) imply that the Jacobian matrix J is symplectic. Indeed, it is immediate to verify that, for any transformation, the matrix J IJ T has an l × l block representation J IJ
= A B C D , where A, B, C, D have as entries A ij = {P i , P j }, B ij = {P i , Q
j }, C
ij = {Q i , P
j }, D
ij = {Q i , Q
j }. Note that if l = 1, then {Q, P } = det J and equations (10.134) reduce to det J = 1. The formal properties of the Poisson brackets will be summarised at the end of the next section. 10.6 Lie derivatives and commutators D efinition 10.16 A Lie derivative associates with the vector field v the differentiation operator L v = N i =1 v i ∂ ∂x i . (10.135)
Evidently the Lie derivative is a linear operator and it satisfies the Leibniz formula: if f and g are two functions on R N with values in R then L v (f g) = f L v g + gL
v f. (10.136) Consider the differential equation ˙x = v(x), (10.137) associated with the field v, and denote by g t (x
) the solution passing through x 0 at time t = 0, i.e. the flow associated with v. The main property of the Lie derivative is given by the following proposition. This proposition also justifies the name ‘derivative along the vector field v’ that is sometimes used for L v . P roposition 10.5 The Lie derivative of a function f : R N → R is given by (L v f )(x) = d dt f ◦ g t (x) | t =0 . (10.138)
10.6 Analytical mechanics: canonical formalism 375 Proof
This fact is of immediate verification: since g t (x) is the solution of (10.137) passing through x for t = 0, d dt t =0 f ◦ g t (x) = N i =1 ∂f ∂x i ˙g 0 i (x) = N i =1 ∂f ∂x i v i (x) = v · ∇f.
(10.139) From the previous proposition it follows that a function f (x) is a first integral of the motion for the flow g t associated with the equation (10.137) if and only if its Lie derivative is zero. If v =
I∇ x H is a Hamiltonian field, then, as we saw, L v f =
{f, H}. Suppose now that two vector fields v 1 and v
2 are given, and denote by g t 1
s 2 the respective flows. In general, the flows of two vector fields do not commute, and hence
g t 1 g s 2 (x) = / g
s 2 g t 1 (x). Example 10.27 Consider the flows g t
(x) = (x 1 cos t − x 2 sin t, x 1 sin t + x 2 cos t),
g t 2 (x) = (x 1 + t, x 2 ), associated with the two vector fields in R 2 given by
v 1 (x) = ( −x 2 , x 1 ), v 2 (x) = (1, 0). One can immediately verify in this case that they do not commute (Fig. 10.3). In addition, the function f 1 (x
, x 2 ) = 1 2 (x 2 1 + x 2 2 ), such that I∇f 1 = v 1 , is a first integral of the motion for g t 1 , and its Lie derivative is L v 1 f 1 = 0, while it is not constant along g t 2 and L v 2 f 1 = / 0. By symmetry, for f 2 (x 1 , x
2 ) =
−x 2 , such that I∇f 2 = v 2 , we have L v 2
2 = 0 and L v 1
2 = / 0. Using the Lie derivative it is possible to measure the degree of non- commutativity of two flows. To this end, we consider any regular function f , defined on R N and we compare the values it assumes at the points g t 1 g s 2 (x) and g s 2 g t 1 (x). The lack of commutativity is measured by the difference ( ∆ f )(t, s, x) = f (g s 2 g t 1 (x)) − f(g
t 1 g s 2 (x)). (10.140) Clearly ( ∆ f )(0, 0, x) ≡ 0 and it is easy to check that the first non-zero term (with starting-point s = t = 0) in the Taylor series expansion of ∆ f with respect to s and t is given by ∂ 2 ( ∆ f ) ∂t∂s (0, 0, x)st, 376 Analytical mechanics: canonical formalism 10.6
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