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i (p(t), q(t), t), ξ i (p(t), q(t), t) in the direct Hamiltonian flow, along which the functions η i , ξ i are all constant. The Poisson brackets are again zero. It is not difficult to check that {Q 1 , P i } = {t, P i } = −
1 ∂ξ 1 /∂t {ξ 1 , η i } = 0, for i > 1, and similarly {Q 1 , Q i } = {t, Q i } = −
1 ∂ξ 1 /∂t {ξ 1 , ξ i } = 0. We must finally check that {Q i , P j } = δ ij for i, j = / 1. Proceeding as usual, we find
{Q i , P j } = {ξ
i , η
j } −
1 ∂ξ 1 /∂t ∂ξ i ∂t {ξ 1 , η j } − ∂η j ∂t {ξ 1 , ξ i } = δ
ij , i, j = / 1. This concludes the proof of the theorem. Remark 10.31 Symplectic rectification leads to a pair of conjugate variables taking respect- ively the values of the Hamiltonian (constant) and of time, while (for l > 1) the remaining coordinates are all first integrals of the motion. Example 10.30 We seek a symplectic rectification of the system with Hamiltonian f (p, q) = p 1 q 2 − p
2 q 1 . Considering directly the equations of the retrograde motion, with Hamiltonian −f(η, ξ) = −η 1 ξ 2 + η
2 ξ 1 , and initial values η i (0) = p i , ξ
i (0) = q
i , i = 1, 2, we find ξ 1
1 cos t
− q 2 sin t, ξ 2 (p, q, t) = q 2 cos t + q 1 sin t,
η 1 (p, q, t) = p 1 cos t
− p 2 sin t, η 2 (p, q, t) = p 2 cos t + p 1 sin t.
384 Analytical mechanics: canonical formalism 10.8 Assuming, for example, that q 2 = / 0, we find that the transformation we seek (setting ξ 1 = 0) is P 1 = p 1 q 2 − p 2 q 1 , Q 1 = arccot
q 1 q 2 , P 2 = p
2 cos
arccot q 1 q 2 + p 1 sin
arccot q 1 q 2 , Q 2 = q 2 cos
arccot q 1 q 2 + q 1 sin
arccot q 1 q 2 . Therefore, possible expressions for P 2 , Q 2 are
P 2 = p 1 q 1 + p
2 q 2 q 2 1 + q 2 2 , Q 2 = q 2 1 + q
2 2 . The new coordinates characterise the first integrals P 1 (constant Hamiltonian), P 2 , Q 2 . Since the motion generated in the space (p, q) is a uniform rotation in the plane q 1 , q 2 together with a uniform rotation in the plane p 1 , p
2 , it is clear that q 2
+ q 2 2 = R 2 = constant (and also p 2 1 + p 2 2 ), while P 2 = constant is equivalent to p · q = constant. 10.8 Infinitesimal and near-to-identity canonical transformations. Lie series The canonical transformations that are ‘near’ (in a sense to be made precise) to the identity transformation have great importance. Indeed, as we shall see in Chapter 12 when we treat the canonical theory of perturbations, using these nearly identical transformations one can study the dynamics of many interesting mechanical systems. For most applications, we only use completely canonical transformations ‘near to identity’. Hence in this and the following sections we study only time-independent transformations. Due to Remark 10.20 this is not a real restriction. D efinition 10.19 Let f and g be two functions of class C ∞ defined on an open set A ⊂ R
2l , with values in R l . Consider ε ∈ R, |ε| 1. An infinitesimal coordinate transformation can be expressed as p = P + εf (P, Q), q = Q + εg(P, Q). (10.166)
T heorem 10.21 If ε is sufficiently small, then the transformation defined by (10.166) is invertible, i.e. for every open bounded subset C of A, with C ⊂ A,
there exists ε 0 > 0 such that for every ε ∈ R, |ε| < ε 0 , the transformation (10.166) 10.8 Analytical mechanics: canonical formalism 385 restricted to C is invertible. The inverse transformation is given, to first order in ε, by P = p
− εf(p, q) + O(ε 2 ), Q = q − εg(p, q) + O(ε 2 ).
Proof The Jacobian matrix of the transformation (10.166) is ∂(p, q) ∂(P, Q)
= 1 + ε ∇ P f ∇ Q f ∇ P g ∇ Q g , where 1 indicates the 2l × 2l identity matrix. Since f and g are in C ∞ , their first derivatives are uniformly bounded on each compact subset of A. Therefore there exists a constant M > 0 such that det ∂(p, q)
∂(P, Q) > 1
− εM on C. It follows that if |ε| < ε 0
transformation is invertible. Since in addition f (P, Q) = f (p, q) + O(ε), g(P, Q) = g(p, q) + O( ), from (10.166) we can immediately deduce (10.167). D efinition 10.20 An infinitesimal transformation (10.166) defines a canonical infinitesimal transformation if {p i , p j } = {q i , q
j } = O(ε
2 ), {q i , p
j } = δ
ij + O(ε 2 ), (10.168) where i, j = 1, . . . , l, and the Poisson brackets are computed with respect to the variables (P, Q). The infinitesimal canonical transformations are the transformations which preserve the fundamental Poisson brackets, up to terms of order O(ε 2
T heorem 10.22 The infinitesimal transformation (10.166) is canonical if and only if there exists a function K : A → R of class C ∞ such that f i (P, Q) = − ∂K ∂Q i , g i (P, Q) =
∂K ∂P i , (10.169)
where i = 1, . . . , l. We say that K is the Hamiltonian associated with the infinitesimal canonical transformation (10.166). 386 Analytical mechanics: canonical formalism 10.8 Proof
The Jacobian matrix of the system (10.166) is ˜ J = 1 + εJ , where J = ∇
f ∇ Q f ∇ P g ∇ Q g is the matrix ∇ X
that the transformation is canonical can be written as (1 + εJ
T ) I(1 + εJ) = I + O(ε 2 ), which is equivalent to J T I + IJ = 0, and hence to the fact that J is Hamiltonian (Definition 10.1). It follows from Theorem 10.5 that the transformation is canonical if and only if the field (f , g) is Hamiltonian, i.e. if it is generated by a Hamiltonian K(P, Q). Hamiltonian matrices are sometimes also called infinitesimally symplectic matrices. This is due to the property just seen, that if J is Hamiltonian then 1 + εJ is symplectic to first order in ε. Due to (10.169) we note that, by inter- preting ε as an infinitesimal ‘time’, the transformation (10.166) is a canonical infinitesimal transformation if and only if (up to terms of order O(ε 2
structure of a Hamiltonian flow with respect to the parameter ε. Example 10.31 The infinitesimal transformation q i = Q i + εQ i , p i = P
i − εP
i , i = 1, . . . , l (10.170) is canonical. Indeed {q i
j } = {p
i , p
j } = 0 and {q i
j } = {Q
i + εQ
i , P
j − εP
j } = {Q
i , P
j } − ε
2 {Q i , P j } = δ ij (1 − ε 2 ). The inverse transformation is Q i = q i 1 + ε = q i − εq i + O(ε 2 ), P i = p i 1 − ε = p i + εp i + O(ε 2 ), and the function K is K = l i =1 P i Q i . Example 10.32 The infinitesimal transformation p = P + 2εQ(1 + cos P ), q = Q + εQ 2 sin P (10.171) is canonical. Indeed, {q, p} = (1 + 2εQ sin P )(1 − 2εQ sin P ) − ε(Q 2 cos P )2ε(1 + cos P ) = 1 − 2ε
2 Q 2 (1 + cos P + sin 2 P ), 10.8 Analytical mechanics: canonical formalism 387 and since l = 1, the Poisson bracket {q, p} is equal to the Jacobian determinant of the transformation. Therefore, if (P, Q) ∈ C, where C is the rectangle (−π, π) × ( −1, 1), the condition ε < 1/ √ 6 is sufficient to ensure the invertibility of the transformation. Evidently the associated Hamiltonian is K =
−Q 2 (1 + cos P ). (10.172) While the infinitesimal canonical transformations are canonical in the approx- imation O(ε
2 ), the near-to-identity canonical transformations which we are about to define depend on a small parameter ε, but are exactly canonical as ε varies. D efinition 10.21 A one-parameter family of completely canonical transforma- tions x = x(X, ε) from the variables x = (p, q) to X = (P, Q) is near to identity if it has the form p = P + εf (P, Q, ε), q = Q + εg(P, Q, ε), (10.173) where ε is a parameter that varies in an open interval I = ( −ε 0
0 ), with
0 < ε 0 1, the functions f , g, A × I → R l are of class C ∞ in all their arguments, and A is an open set in R 2l . T heorem 10.23 Let C be a compact subset of R 2l . Every near-to-identity canon- ical transformation defined in an open neighbourhood of C admits a generating function F (q, P, ε) of the form F (q, P, ε) = l i =1 q i P i + ε F(q, P, ε), (10.174)
and vice versa. Here F is a function of class C ∞ in all its arguments, for every ε ∈ (−ε
0 , ε
0 ), where ε 0 is a sufficiently small positive constant. Proof From the second of equations (10.173) it follows that ∂q i
j = δ
ij + ε
∂g i ∂Q j , and therefore as (P, Q) ∈ C varies, the condition (10.121), i.e. det(∇ Q q) = / 0, is certainly satisfied, if ε 0 is sufficiently small. Hence there exists a regular function Q(q, P, ε) such that Q = q + ε Q(q, P, ε). Substituting it into the first of equations (10.173) we find p = P + ε P(q, P, ε), 388 Analytical mechanics: canonical formalism 10.8 where
P(q, P, ε) = f(P, q+εQ(q, P, ε), ε). We now recall that if the transformation (10.173) is canonical, the form l i
(p i dq i + Q
i dP i ) can be integrated to find the generating function (10.175) (see (10.117)). Conversely, since (q, P) varies in a compact subset of R 2l , if ε 0 is sufficiently small then det
∂ 2 F ∂q i ∂P j = det
δ ij + ε ∂ 2 F ∂q i ∂P j > 0,
and the equations p =
∇ q F = P + ε ∇ q F, Q = ∇ P F = q + ε ∇ P
(10.175) generate a near-to-identity canonical transformation. Example 10.33 Consider the function F (q, P, ε) = qP + εq 2 (1 + cos P ). (10.176) Since ∂
2 F/∂q∂P = 1 − 2εq sin P , as (q, P ) varies in a compact subset, if |ε| < ε 0 is sufficiently small, the function F generates a near-to-identity canonical transformation. For example if (q, P ) ∈ [−a, a] × [−π, π] we set ε 0
transformation generated by F is defined implicitly by Q = q
− εq 2 sin P, p = P + 2εq(1 + cos P ). Solving the first equation for q we find q = 1
1 − 1 − 4εQ sin P , where the choice of negative sign for the determination of the square root is fixed by the requirement that q → Q when ε → 0. Taking the Taylor series of the square root: − √
− x = −1 + x 2 + ∞ n =2 (2n
− 3)!! 2 n n! x n , and taking into account that x = 4εQ sin P we find q = Q + εQ 2 sin P + ∞ j =2 (2j − 1)!!
(j + 1)! 2 j ε j (sin P ) j Q j +1 . (10.177) The same result is obtained by an application of Lagrange’s formula (see Theorem 5.5): q = Q + εQ 2 sin P + ∞ j =2 ε j j! (sin P ) j d j −1 dQ j −1 Q 2j ,
10.8 Analytical mechanics: canonical formalism 389 considering that d j −1 dQ j −1 Q 2j = 2 j (2j − 1)!! j + 1
Q j +1 , as is easily verified by induction. Substituting the expression for q into the expression for p we arrive at p = P + 2ε(1 + cos P ) Q + εQ 2
∞ j =2 (2j − 1)!!
(j + 1)! 2 j ε j (sin P ) j Q j +1 . (10.178) The comparison between Examples 10.32 and 10.33 sheds light on the differ- ence between infinitesimal canonical transformations and near-identical canonical transformations. Clearly, the transformation (10.171) coincides with equations (10.177), (10.178) up to terms of order O(ε 2
with (10.171) is K = −Q 2 (1 + cos P ), comparing this with (10.176) suggests that by setting F(q, P) = −K(q, P) in Theorem 10.17 we obtain the generating function of a near-to-identity canonical transformation starting from an infinites- imal canonical transformation. This is precisely the conclusion of the following theorem.
T heorem 10.24 To every infinitesimal canonical transformation (10.166) with associated Hamiltonian K (see (10.169)) there corresponds a near-to-identity canonical transformation. The latter coincides with (10.166) up to terms of order O(ε
2 ). The transformation can be obtained starting from the generating function (10.174) by setting F = −K(q, P). Conversely, to every near-to-identity canonical transformation (10.173) there corresponds an infinitesimal canonical transform- ation (10.166) obtained by neglecting terms of order O(ε 2
associated Hamiltonian is given by K = −F(Q, P, 0). Proof To prove the first statement it is enough to note that from equation (10.176), setting F = −K(q, P), it follows that p = P − ε∇
q K, Q = q − ε∇ P K, and hence p = P
− ε∇ Q K + O(ε 2 ), q = Q + ε ∇ P K. The second part of the theorem has an analogous proof. 390 Analytical mechanics: canonical formalism 10.8 We saw in Theorem 10.13 that the Hamiltonian flow is canonical. Considering time as a parameter and setting ε = t in an interval ( −ε 0 , ε 0 ), this flow gives an example of a near-to-identity canonical transformation, while neglecting terms of order
O(ε 2 ) it provides an example of an infinitesimal canonical transformation. Indeed, consider the canonical equations (10.90) for a Hamiltonian H(p, q), where (P, Q) denote the initial conditions (at time t = ε = 0) and (p, q) denote the solutions of (10.90) at time t = ε. An integration of equations (10.90) that is accurate to first order in ε yields p = P − ε∇
Q H(P, Q) + O(ε 2
q = Q + ε ∇ P H(P, Q) + O(ε
2 ), (10.179) and hence equations (10.169) are satisfied with K = H. We now show how it is always possible, at least in principle, to formally construct a near-to-identity transformation associated with a given Hamiltonian H. Let (p, q) = S t (P, Q) = (p(P, Q, t), q(P, Q, t)) (10.180) be the Hamiltonian flow associated with H. As we saw in Section 10.5, S t defines
an evolution operator U t acting on the observables of the system. If the Hamilto- nian H(p, q) is independent of time, and we consider the action of U t on the functions f (P, Q) ∈ C
∞ (R 2l ), then we have d dt (U t f )(P, Q) = ( {f, H} ◦ S t )(P, Q) = {f, H}(p, q) = (−D H f )(p, q), (10.181) where D H = {H, ·} (see (10.145)) is called an infinitesimal generator of U t . T heorem 10.25 For every t ∈ R we formally have that U t
−tD H , (10.182) i.e.
(U t f )(P, Q) = ∞ j =0 ( −t)
j j! (D j H f )(P, Q) (10.183) as long as the series converges. Remark 10.32 Here D
j H denotes the operator D H applied j times if j ≥ 1, and the iden- tity operator D 0 H
operator U t is called the Lie series. Proof of Theorem 10.25 By the theorem of the existence and uniqueness for ordinary differential equations, the Hamiltonian flow is uniquely determined and it is a one-parameter group
10.8 Analytical mechanics: canonical formalism 391 of diffeomorphisms. Therefore it is sufficient to check that the series (10.183) Download 10.87 Mb. Do'stlaringiz bilan baham: 
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