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E | |u| ds E = 1 }. (A6.33)
In addition, ψ transforms the Hamiltonian flow M E into the geodesic flow, since, as we have seen, s = τ and (A6.24) coincides with (A6.29) (recall that q is uniquely determined by the knowledge of p and of dp/dτ , see (A6.21)). Due to Remark A3.2, it is immediate to compute the Gaussian curvature of the Riemannian manifold (A E , ds 2 E ). Indeed, setting g = 4k 2 ( |p|
2 − 2E)
2 , (A6.34) the Gaussian curvature K of (A E , ds 2 E ) is given by K = − 1 g ∂ ∂p 1 1 √ g ∂ ∂p 1 √ g + ∂ ∂p 2 1 √ g ∂ ∂p 2 √ g = ( |p|
2 − 2E)
2 4k 2 ∂ ∂p 1 2p 1 |p| 2 − 2E
+ ∂ ∂p 2 2p 2 |p| 2 − 2E = − 1 4k 2 [4p 2 1 + 4p 2 2 − 4(|p| 2 − 2E)] = − 2E k
. (A6.35)
If E < 0, the stereographic projection p 1 = k 2 |E| x 1 k/ 2 |E| − x 0 , p 2 = k 2 |E|
x 2 k/ 2 |E| − x
0 (A6.36)
transforms A E into the sphere x 2 0 + x 2 1 + x 2 2 = k 2 2 |E|
(A6.37) without the point x 0 = k/
2 |E|, x
1 = x
2 = 0. The sphere (A6.37) has radius k/ 2
into the first fundamental form (A6.37). If E = 0, setting z = p 1
2 = Re
i ϑ , w = x 1 + ix 2 = re
iϕ , (A6.38) 740 Kepler’s problem, linear oscillators and geodesic flows A6 the inversion z = 2k w (A6.39) transforms A E into R
2 \{(0, 0)} with the Euclidean metric (dx 1 )
+ (dx 2 ) 2 = dw dw = (dr) 2 + r
2 (dϕ)
2 . Finally, for E > 0, with the same notation as in (A6.38), the inversion z = √ 2E w transforms A E into D and (A6.26) into (ds E ) 2 = 4k 2 ( |z| 2 − 2E)
2 dz dz =
4k 2 (2E/ |w| 2 − 2E) 2 2E |w| 4 dw dw
= 4 2E/k 2 dw dw
(1 − |w|
2 ) 2 , (A6.40)
which coincides with (A6.1) by setting R = k/ √ 2E. APPENDIX 7: FOURIER SERIES EXPANSIONS Fourier trigonometric series are a particular case of the more general concept of expansion of any element of a Hilbert space with respect to a complete orthonormal basis. We briefly summarise the essential notions (for more details and proofs, see for example Rudin (1974)). A real (respectively, complex) vector space V is endowed with a norm if to all its elements x we can associate a real number x (its norm) with the following properties: (i)
x ≥ 0, ∀ x ∈ V ; (ii) λx =
|λ| x , ∀ x ∈ V , ∀ λ ∈ R (respectively, ∀ λ ∈ C); (iii)
x + y ≤ x + y , ∀ x, y ∈ V . A sequence {x n } in a normed space V converges in the norm to the limit x ∈ V if lim n →∞ x n − x = 0.
A sequence {x n } in a normed space is a Cauchy sequence if x n − x m → 0
for n, m → +∞.
A normed space is complete if all Cauchy sequences are convergent. A complete normed space is called a Banach space. A Banach space whose norm is generated by a scalar product is called a Hilbert space. If the space is real (respectively, complex) a scalar product (x, y) is a symmetric (respectively, Hermitian) positive definite bilinear form. It is easily seen that (x, x) 1/2 has the properties of a norm. In a Hilbert space we then set x 2 = (x, x). A complete orthonormal basis {e i
that (e i , e j ) = δ
ij and (x, e i ) = 0,
∀ i ⇒ x = 0. T heorem A7.1 Let {e i } be a complete orthonormal basis of a Hilbert space H. Then every element x ∈ H can be uniquely represented by the series x = i
i e i . (A7.1)
The coefficients α i are called Fourier coefficients and are defined by α i = (x, e i ). (A7.2) The Parseval identity x 2 = i α 2 i (A7.3) holds. 742 Fourier series expansions A7 T
it is separable. C orollary A7.1 Every separable real (respectively, complex) Hilbert space (infinite dimensional) is isomorphic to the space 2 of sequences (α i ) i ∈N in R
(respectively, in C) for which the series of i |α i | 2 is convergent with respect to the scalar product (x, y) = ∞ i
x i y i (respectively (x, y) = ∞ i
x i y i ). Consider the Hilbert space L 2 ((0, 2π), R) of measurable functions v : (0, 2π) → R which are square integrable, with the scalar product (u, v) = 2π 0 u(x)v(x) dx. A complete orthonormal basis for it is given by the functions 1/ √ 2π, (1/ √ π) sin nx, (1/ √
∈ N, n > 0. Every measurable function V : R → R, 2π- periodic (hence such that V (x + 2π) = V (x) for every x ∈ R), such that its restriction v = V | (0,2π)
is also square integrable, determines uniquely an element v ∈ L 2 ((0, 2π), R) which can be written as v(x) = a 0 + ∞ n =1 [a n cos(nx) + b n sin(nx)], (A7.4) where
a 0 = 1 √ 2π 2π 0 v(x) dx, a n = 1 √ π 2π 0 v(x) cos(nx) dx, b n =
√ π 2π 0 v(x) sin(nx) dx. (A7.5) Equations (A7.4) and (A7.5) take a particularly compact form if we express sine and cosine through complex exponentials: v(x) =
k ∈Z ˆ v k e ikx , ˆ v k = 1 √ 2π 2π 0 v(x)e −ikx dx.
(A7.6) The relation between the coefficients (ˆ v k
k ∈Z and (a n , b
n ) n ∈N is given by ˆ v
= a 0 , ˆ v k = 1 2 (a k − ib
k ) if k
≥ 1, ˆv k = 1 2 (a −k + ib
−k ) = ˆ
v ∗ −k if k ≤ −1.
(A7.7) Now let v : R l → R be a measurable function, periodic of period 2π in each of its arguments: v(x
1 + 2π, . . . , x l ) = v(x
1 , x
2 + 2π, . . . , x l ) = . . . = v(x 1 , x 2 , . . . , x l + 2π)
= v(x 1 , . . . , x l ). (A7.8) Assume also that its restriction v to (0, 2π) l is square integrable. A7 Fourier series expansions 743 The function v is hence defined on the l-dimensional torus T l = (R/2πZ) l ,
v(x) = m∈Z
l ˆ v m e im·x . (A7.9)
The coefficients ˆ v m of the series expansion (A7.9) are determined as follows: ˆ v m = 1 (2π) l T l v(x)e
−im·x dx.
(A7.10) If the function v is regular, its Fourier coefficients belong to 2 (Z
, C) and decay at infinity at a rate related to the degree of regularity of the function. T heorem A7.3 If v is of class C r , there exists a constant M > 0 such that for every m ∈ Z
l , m =
/ 0 we have |ˆv
m | ≤ M(|m
1 | + · · · + |m l |)
. (A7.11)
If v is of class C ∞ , for every positive integer r there exists a constant M > 0 such that for every m ∈ Z l
/ 0 we have |ˆv
m | ≤ M(|m
1 | + · · · + |m l |)
. (A7.12)
If v is analytic, there exist two constants M > 0 and δ > 0 such that for every m ∈ Z l we have
|ˆv m | ≤ Me −δ(|m 1 |+···+|m l |) . (A7.13) Proof
For simplicity we only prove the estimates (A7.11) and (A7.12). For the proof of (A7.13) see for example Sternberg (1969) and Rudin (1974). We first remark that (A7.12) is an obvious consequence of (A7.11). Let l = 1. The proof of (A7.11) depends on the identity e −imx
= 1 −im d dx e −imx , m = / 0. (A7.14)
From (A7.10) it follows that ˆ v m = 1 2π 2π 0 v(x)e −imx
dx = 1 2π 2π 0 v(x) 1 −im
d dx e −imx dx,
and hence, integrating by parts, we find ˆ v m = 1 2π 1 ( −im) v(x)e
−imx x =2π x =0 + 1 i2πm
2π 0 v (x)e −imx dx.
744 Fourier series expansions A7 The first term is zero due to the periodicity of v(x)e −imx . If v is of class C r
can iterate this procedure r times to obtain ˆ v m = 1 2π(im) r 2π 0 v (r) (x)e −imx
dx, from which it immediately follows that |ˆv m
1 2π |m| r 2π 0 |v (r)
(x) | dx ≤
M |m|
r , where M = max 0≤x≤2π |v (r) (x) |. The existence of M is guaranteed by the assumption that v is of class C r . If l
≥ 1, it is sufficient to observe the following. Since m = / 0, there exists at least one component m j = / 0. From (A7.10), integrating by parts, we find ˆ v m = 1 (2π) l T l v(x)
− 1 im j ∂ ∂x j e −im·x dx = − 1 im j 1 (2π)
l T l ∂v ∂x j (x)e −im·x
dx, from which, iterating this procedure r times, we obtain |ˆv m
˜ M (max 1≤j≤l |m j |) r , (A7.15) where ˜
M = max 1≤j≤l
max x∈T
l ∂ r v/∂x r j (x) . On the other hand, max 1≤j≤l |m
| ≤ |m 1 | + · · · + |m l | ≤ l max 1≤j≤l |m
|. Substituting this into (A7.15) yields the desired estimate (A7.11), with M = l r
M . APPENDIX 8: MOMENTS OF THE GAUSSIAN DISTRIBUTION AND THE EULER Γ FUNCTION
The moments of the Gaussian distribution are the integrals of the type µ n = +∞ −∞ x n e −ax 2 dx, (A8.1) with a being a positive constant and n ∈ N. P
n of the Gaussian distribution are µ 0
π a , µ 2n+1
= 0, for every n ≥ 0, (A8.2)
µ 2n = (2n − 1)!! π a (2a) −n , for every n ≥ 0.
Proof The proof is a direct computation. First of all, consider µ 2 0
µ 2 0 = +∞ −∞ e −ax
2 dx 2 = +∞ −∞ e −ax
2 dx +∞ −∞ e −ay 2 dy = +∞ −∞ dx +∞ −∞ e −a(x 2 +y 2 ) dy = 2π 0 dϕ +∞ 0 re −ar 2 dr = π a , where we have used the substitution x = r cos ϕ, y = r sin ϕ. Evidently µ 2n+1 = 0, because the integrand is an odd function of x. To compute µ 2n we first note that x 2n e −ax 2 = ( −1) n ∂ n ∂a n e −ax
2 , From which it immediately follows that µ 2n = ( −1) n d n da n +∞ −∞ e −ax 2 dx = √ π( −1) n − 1 2 − 3 2 · · · − 2n − 1
2 a −(2n+1)/2 = (2n − 1)!!
π a (2a) −n .
746 Moments of the Gaussian distribution and the Euler Γ function
A8 It is often also useful to compute the integral between 0 and ∞. For integrals with even n it is enough to divide the previous result by two. For those with odd n, we can check with a sequence of integration by parts that ∞ 0 x 2n+1
e −x 2 dx = 1 2 n!, (A8.3)
and hence ∞ 0 x 2n+1
e −ax
2 dx =
1 2 a −(n+1) n!.
(A8.4) There exists an obvious relation between the moments of the Gaussian distribution and the Euler Γ function Γ (z) =
∞ 0 t z −1 e −t dt,
z > 0. (A8.5)
Indeed, after the substitution ax 2 = t we immediately find ∞ 0 x n e −ax 2 dx =
1 2 a −(n+1)/2 Γ n + 1 2 . (A8.6) For the function Γ we have Γ (n + 1) = n!, ∀ n ∈ N. (A8.7)
This can be deduced by induction from equation (A8.10) below, while equation (A8.2) yields Γ 1
= √ π, Γ k +
1 2 = 2 −k (2k
− 1)! √ π. (A8.8) Another interesting formula is Γ (z)
Γ (1 − z) = π sin zπ
, 0 < z < 1, (A8.9) illustrating that Γ (z) diverges for z → 0+ as 1/z. Equation (A8.9) is important because, by using recursively the property Γ (z + 1) = z Γ (z),
(A8.10) we can reduce to the computation of Γ only for z ∈ (0, 1) and, due to (A8.9), for z
∈ 0, 1 2 . We can also see that Γ (z) has a unique minimum for z 1.4616 . . .. For z
1 the famous Stirling formula holds: Γ (z + 1) = √ 2πz z
z e −z e α (z)/12z , with α
∈ (0, 1), (A8.11)
A8 Moments of the Gaussian distribution and the Euler Γ function
747 and the approximation Γ (z + 1)
√ 2πz
z e z (A8.12) can be used when z(log z −1) 1 (e.g. for z = 10 we have z(log z −1) 15.4 and
the relative error in (A8.12) is less than 1%, while for z = 50, z(log z − 1)
1680 and the relative error is about 0.2%). Bibliography (a) Books Abraham R., Marsden J.E. (1978). Foundations of mechanics, Benjamin Cum- mings, Reading, MA. Agostinelli C., Pignedoli A. (1989). Meccanica analitica, Mucchi, Modena. Amann H. (1990). Ordinary differential equations, de Gruyter Studies in Mathem- atics, Vol. 13, W. de Gruyter, Berlin. Arnol’d V.I. (1978a). Mathematical methods of classical mechanics, Springer-Verlag, New York. —— (1978b). Ordinary differential equations, MIT Press, Boston, MA. —— (1983). Geometric methods in the theory of ordinary differential equations, Springer-Verlag, New York. —— (1990). Huygens and Barrow, Newton and Hooke, Birkh¨ auser Verlag, Boston. —— (1991). Theory of singularities and its applications (Lezioui Fermiane), Cam- bridge University Press. Arnol’d V.I., Avez A. (1968). Ergodic problems of classical mechanics, Benjamin, New York. Arnol’d V.I., Kozlov V.V., Neishtadt A.I. (1988). Dynamical systems III, Encyclo- pedia of Mathematical Sciences, Springer-Verlag, Berlin. Arrowsmith D.K., Place C.M. (1990). An introduction to dynamical systems, Cambridge University Press. Barrow-Green J. (1997). Poincar´ e and the three body problem, History of Math- ematics 11, American Mathematical Society and London Mathematical Society, Providence, RI. Bedford T., Keane M., Series C. (1991). Ergodic theory, symbolic dynamics and hyperbolic spaces, Oxford University Press. Beletski V. (1986). Essais sur le mouvement des corps cosmiques, Mir, Moscow. Benettin G., Galgani L., Giorgilli A. (1991). Appunti di meccanica razionale, CUSL, Milano. Bertin G. (2000). Dynamics of galaxies, Cambridge University Press. Binney J., Tremaine S. (1987). Galactic dynamics, Princeton University Press. Download 10.87 Mb. Do'stlaringiz bilan baham: 
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