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solves equation (10.181). Formally, this has an immediate verification: indeed, by differentiating the series (10.183) term by term we find d dt
j =0 ( −t) j j! (D j H f )(P, Q) = − ∞ j =1 ( −t)
j −1 (j − 1)! (D j H f )(P, Q) = −D
∞ j =1 ( −t)
j −1 (j − 1)! (D j −1 H f )(P, Q) = −D H ∞ n =0 ( −t)
n n! (D n H f )(P, Q) = −D H (e −tD
H f )(P, Q) = {f, H}(p, q), where we set n = j − 1. Example 10.34 Let l = 1, H(q, p) = qp. The Hamiltonian flow is clearly given by p = e
−t P, q = e t Q. Consider the functions f 1 (p, q) = q and f 2 (p, q) = p and apply equations (10.183), to obtain p = U
t f 2 = ∞ j =0 ( −t) j j! (D j H p) | (p,q)=(P,Q) , q = U
t f 1 = ∞ j =0 ( −t) j j! (D j H q) | (p,q)=(P,Q) . On the other hand, −D H q = {q, H} = q, D 2 H q = −D H ( −D H q) = q, and hence ( −D H ) j q = q for every j ≥ 1. In addition (−D H )
p = ( −1)
j p, and substituting into the series we find p = P
∞ j =0 ( −t)
j j! = e −t P, q = Q ∞ j =0 t j j! = e t Q. Example 10.35 Let l = 1, H(p, q) = (q 2 + p
2 )/2. The associated Hamiltonian flow is p = −Q sin t + P cos t, q = Q cos t + P sin t. 392 Analytical mechanics: canonical formalism 10.8 Applying (10.183) to q and p, and observing that −D H q = p, D 2 H q = −D H p = −q, −D 3 H q = =
−D 2 H p = −p and D
4 H q = −D 3 H p = q, we find p =
∞ j =0 t j j! ( −D H ) j p = P ∞ j =0 ( −1)
j t 2j (2j)! − Q
∞ j =0 ( −1)
j t 2j+1 (2j + 1)! = P cos t − Q sin t, q =
∞ j =0 t j j! ( −D H ) j q = Q ∞ j =0 ( −1)
j t 2j (2j)! + P
∞ j =0 ( −1)
j t 2j+1 (2j + 1)! = Q cos t + P sin t. For example, if we consider a function f (p, q) = qp, on the one hand, we have (U t f )(P, Q) = (Q cos t + P sin t)( −Q sin t + P cos t) = (P 2
2 ) sin t cos t + P Q(cos 2 t
2 t),
while, on the other, from −D H f = {qp, H} = p 2 − q
2 , D
2 H f = −4pq and from equation (10.183) it follows that (U t
∞ j =0 t j j! ( −D H ) j f = P Q + t(P 2 − Q
2 ) + t 2 2 ( −4P Q) +
t 3 3! ( −4) P
2 − Q
2 + · · · . This coincides with the
series expansion of (P
− Q 2 ) sin t cos t + P Q (cos 2 t − sin 2 t). Example 10.36 Let l = 1, H = p 2 /2. Then
p = P, q = P t + Q. If f (p, q) = p n q m , with n and m non-negative integers, then (U t f )(P, Q) = (P t + Q) m P n . On the other hand, ( −D H
j f = m(m
− 1) . . . (m − j + 1)p n +j q m −j for j = 1, . . . , m, and ( −D H ) j f = 0 for all j > m. Applying equations (10.183) we find (U t f )(P, Q) = m j =0 t j j! m(m
− 1) . . . (m − j + 1)P n +j Q m −j = P n m j =0 m j t j P j Q m −j = P n (P t + Q) m (in the last equality we used Newton’s binomial formula). Example 10.37 Let H = (q 2 p
)/2. Since pq = √ 2E is constant, the canonical equations can immediately be integrated: p = P e
−P Qt , q = Qe P Qt .
10.9 Analytical mechanics: canonical formalism 393 On the other hand, ( −D H ) j p = (
−1) j q j p j +1 , (
−D H ) j q = p
j q j +1 , and hence by equations (10.183) we have p =
∞ j =0 ( −t)
j j! Q j P j +1 = P e
−P Qt , q = ∞ j =0 t j j! Q j +1 P j = Qe P Qt . 10.9 Symmetries and first integrals In this section we briefly consider the relations between the invariance properties of the Hamiltonian for groups of canonical transformations and the first integrals. For a more detailed study of this important topic in analytical mechanics, see Arnol’d (1979a) (Appendix 5). Let H : R 2l → R be a regular Hamiltonian. D efinition 10.22 A completely canonical transformation x = ˜x(X) of R 2l is a
symmetry of H if the Hamiltonian is invariant for the transformation, and hence if H(˜ x(X)) = H(X). (10.184)
Example 10.38 If H has one cyclic coordinate x i (note that x i can be either a coordinate q or a kinetic momentum p), H is invariant for the translations x i → x i + α.
Example 10.39 The rotations around the origin in R 2 :
q = −P sin α + Q cos α, are a symmetry of H = (p 2 + q 2 )/2.
Another class of interesting examples is given by the following proposition. P roposition 10.7 If H(p, q) is the Legendre transform of the Lagrangian L(q, ˙q), and the point transformation q = ˜ q(Q) is admissible for L (see Definition 9.1), the associated completely canonical transformation q = ˜
q(Q), p = (J
T (Q))
−1 P, (10.185) where J = (J ij ) = (∂ ˜ q i /∂Q j ), is a symmetry of the Hamiltonian H. 394 Analytical mechanics: canonical formalism 10.9 Proof
In the new variables the Hamiltonian ˆ H(P, Q) is obtained as the transform of H(p, q) and it is also the Legendre transform of the Lagrangian ˜ L(Q, ˙
Q) = L(˜
q(Q), J (Q) ˙ Q), i.e.
ˆ H(P, Q) = H((J T )
P, ˜ q(Q)) = P · ˙Q − ˜L(Q, ˙Q). We now satisfy the hypothesis ˜ L(Q, ˙ Q) = L(Q, ˙ Q); hence to compare the new with the old Hamiltonian we must compare P · ˙Q with p · ˙q. We already know that they take the same values (see Section 10.2), but we want to see that if p · ˙q = F (p, q) then P · ˙Q = F (P, Q). Obviously it is enough to show that if ˙q = f (p, q) then ˙ Q = f (P, Q). This holds because ˙q = f (p, q) can be obtained by inverting the system p = ∇ ˙ q L(q, ˙q). Because of the admissibility of the tramsformation, this sytem is formally identical to P = ∇ ˙ Q L(Q, ˙
Q). In conclusion ˆ H(P, Q) = H(P, Q). D efinition 10.23 A one-parameter family s ∈ R of completely canonical trans- formations x = ˜ x(X, s) of R 2l is called a one-parameter group (of completely canonical transformations) if it possesses the following properties: (1) ˜
x(X, 0) = X for all X ∈ R
2l ; (2) ˜ x(˜ x(X, s
1 ), s
2 ) = ˜
x(X, s 1 + s 2 ) for every s 1 , s
2 ∈ R and for every X ∈ R 2l .
∈ R the transformation ˜x(X, s) is a symmetry of H, the group is a one-parameter group of symmetries of H. Remark 10.33 For the groups of point transformations see Problem 9 of Section 10.14. We now examine how it is possible to interpret any one-parameter group of completely canonical transformations as a Hamiltonian flow. D efinition 10.24 Let ˜x(X, s) be a one-parameter group of completely canonical transformations of R 2l . The vector field v(x) = ∂ ˜
x ∂s (x, 0) (10.186) is called an infinitesimal generator of the group of transformations. The following theorem clarifies the role of the infinitesimal generator. T heorem 10.26 The infinitesimal generator v(x) of a one-parameter group ˜ x(X, s) of completely canonical transformations is a Hamiltonian field. In addition the group of transformations coincides with the corresponding Hamiltonian flow, and hence it is a solution of the system ˙x(t) = v(x(t)), x(0) = X. (10.187)
10.10 Analytical mechanics: canonical formalism 395 Proof
We first check that ˜ x(X, t) is a solution of equations (10.187). Because of the group properties we have, setting x(t) = ˜ x(X, t), that ˙x(t) = lim ∆ t→0
˜ x(X, t +
∆ t) − ˜x(X, t) ∆ t = lim ∆ t→0 ˜ x(x(t), ∆ t) − ˜x(x(t), 0) ∆ t = v(x(t)). Since by the hypothesis, the Jacobian matrix J = ∇ X ˜ x(X, t) is symplectic for every t, we deduce by Lemma 10.1 that the matrix B = (∂J /∂t)J −1 is Hamiltonian. Now note that we can write ∂J ∂t = ∂ ∂t ∇ X ˜ x(X, t) = ∇ X v(˜ x(X, t)) = ( ∇ x
Using the fact that ˜ x(X, t) solves equations (10.187), it follows that ∇ x
= (∂J /∂t)J −1 and hence the field v(x) is Hamiltonian (Theorem 10.5). We can now prove the following extension of Noether’s theorem. Recall how in the Lagrangian formulation (Theorem 4.4) the validity of this theorem was limited to symmetry groups associated to point transformations. T heorem 10.27 (Noether, Hamiltonian formulation) If a system with Hamilto- nian H(x) has a one-parameter group of symmetries ˜ x(X, t), the Hamiltonian K(x) of which the group is the flow is a first integral for the flow associated with H.
Proof The invariance of H can be interpreted as its being constant along the flow generated by K. Therefore L v H = {H, K} = 0. Conversely this implies that K is a first integral for the flow generated by H. In summary, if f (x), g(x) are in involution, recalling Remark 10.30 we see that: (i) the Hamiltonian flow generated by f (x) has g(x) as first integral and vice versa ;
(ii) the two flows associated with f and g commute; (iii) the flow generated by f (x) represents a symmetry for the Hamiltonian g(x) and vice versa. 10.10
Integral invariants In this section, which can be omitted at a first reading, we want to character- ise the canonical transformations using the language of differential forms (see Appendix 4). For simplicity, we limit the exposition to the case of differential forms in R 2l , while in the next section we introduce the notion of a symplectic manifold which allows us to extend the Hamiltonian formalism to a wider context. 396 Analytical mechanics: canonical formalism 10.10 T
canonical if and only if l i =1 dp i ∧ dq i = l i =1 dP i ∧ dQ i . (10.188) Remark 10.34 A transformation satisfying (10.188) is also called a symplectic diffeomorphism as it preserves the symplectic 2-form 1 ω = l i =1 dp i ∧ dq i . (10.189) Proof of Theorem 10.28 The proof follows from an immediate application of the Lie condition (10.190). From l
=1 p i dq i − l i =1 P i dQ i = df
(10.190) (note that, since the transformation is independent of time, ˜ d = d), if we perform an external differentiation of both sides and take into account d 2 f = 0 we find (10.188). Conversely, since (10.188) is equivalent to d l i =1 p i dq i − l i =1 P i dQ i = 0, (10.191) we immediately deduce (10.190) because of Poincar´ e’s lemma (Theorem 2.2, Appendix 4): every closed form in R 2l is exact. From Theorem 10.188 we easily deduce some interesting corollaries. C orollary 10.7 A canonical transformation preserves the differential 2k-forms: ω 2k = 1≤i 1
2
k
≤l i 1 ∧ . . . ∧ dp i k ∧ dq i 1 ∧ . . . ∧ dq i k , (10.192)
where k = 1, . . . , l. Proof
If the transformation is canonical, it preserves the 2-form ω, and hence it also preserves the external product of ω with itself k times (see (A4.20)): Ω k
∧ . . . ∧ ω = i 1 ,...,i k dp i 1 ∧ dq i 1 ∧ . . . ∧ dp i k ∧ dq i k , (10.193) 1 Be careful: in spite of the same notation, this is not to be confused with the Poincar´ e– Cartan form. 10.11 Analytical mechanics: canonical formalism 397 which is proportional to ω 2k : Ω k = (
−1) k −1 k!ω 2k . (10.194) Evidently Corollary 10.21 for k = l can be stated as follows. C orollary 10.8 A canonical transformation preserves the volume form ω 2l = dp 1 ∧ . . . ∧ dp l ∧ dq
1 ∧ . . . ∧ dq l .
Remark 10.35 The forms ω 2k have a significant geometrical interpretation. If k = 1, the integral of the form ω on a submanifold S of R 2l is equal to the sum of the areas (the sign keeps track of the orientation) of the projections of S onto the planes (p i , q i ). Analogously, the integral of ω 2k is equal to the sum of the measures (with sign) of the projections of S onto all the hyperplanes (p i 1 , . . . , p i k , q i 1 , . . . , q i k ), with 1
≤ i 1
k ≤ l. It follows that a completely canonical transformation preserves the sum of the measures of the projections onto all coordinate planes and hyperplanes (p i 1
i k , q i 1 , . . . , q i k ). 10.11 Symplectic manifolds and Hamiltonian dynamical systems D efinition 10.25 A differentiable manifold M of dimension 2l is a symplectic manifold if there exists a closed non-degenerate differential 2-form 2 ω, i.e. such that dω = 0 and, for every non-zero vector v ∈ T
m M tangent to M at a point m, there exists a vector w ∈ T
m M such that ω(v, w) = / 0. Example 10.40 As seen in Section 10.1, every real vector space of even dimension can be endowed with a symplectic structure which makes it into a symplectic manifold. An example is R 2l with the standard structure ω = l i =1 dp i ∧ dq i . Example 10.41 The tori of even dimension T 2l with the 2-form ω = l i =1 dφ i ∧ dφ i +l are symplectic manifolds. This is an interesting case. Indeed, because of the particular topology (the tori are not simply connected) it is easy to construct examples of vector fields which are locally Hamiltonian, but not globally a Hamiltonian. For example, the infinitesimal generator of the one-parameter group of translations φ i
i + α
i t, where α ∈ R 2l
but not globally, a Hamiltonian vector field, since every function H : T 2l → R necessarily has at least two points where the gradient, and hence the field I∇ φ H, vanishes. 2 Be careful: not to be confused, in spite of the same notation, with the Poincar´ e–Cartan form, which is a 1-form on M × R.
398 Analytical mechanics: canonical formalism 10.11 Example 10.42 A natural Lagrangian system is a mechanical system subject to ideal constraints, frictionless, fixed and holonomous, and subject to conservative forces. Its space of configurations is a differentiable manifold S of dimension l (the number of degrees of freedom of the system) and the Lagrangian L is a real function defined on the tangent bundle of S, L : T S → R. The kinetic energy T (q, ˙q) = 1 2
i,j =1 a ij ˙ q i ˙ q j defines a Riemannian metric on S:(ds) 2 =
i,j =1 a ij q i dq j , and the Lagrangian can be written then as L = 1 2 |ds/dt| 2 − V (q), where V is the potential energy of the conservative forces. The Hamiltonian H is a real function defined on the cotangent bundle M = T ∗ S of the space of configurations H : M → R, through a Legendre transformation applied to the Lagrangian L : H = p · ˙q − L. If A denotes the matrix a ij of the kinetic energy, H(p, q) = 1 2 p · A −1 p + V (q). The Download 10.87 Mb. Do'stlaringiz bilan baham: 
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