When physics helps mathematics: Calculation of the sophisticated multiple integral
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hen Physics Helps Mathematics: Calculation of the Sophisticated Multiple Integral1 A. L. Kholodenkoa and Z. K. Silagadzeb a
H.L .Hunter Laboratories, Clemson University, Clemson, SC 296340973, USA Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk, 630090 Russia
b
Abstract—There exists a remarkable connection between the quantum mechanical LandauZener problem and purely classicalmechanical problem of a ball rolling on a Cornu spiral. This correspondence allows us to calculate a complicated multiple integral, a kind of multidimensional generalization of Fresnel integrals. A direct method of calculation is also considered but found to be successful only in some lowdimensional cases. As a byproduct of this direct method, an interesting new integral representation for ζ(2) is obtained. DOI: 10.1134/S1063779612060068 1
1. INTRODUCTION According to Vladimir Arnold [1], mathematics can be considered as some branch of physics. In writ ing this note we have no intentions to advocate such point of view. Nevertheless, in our opinion the follow ing calculus problem is a hard nut to crack if only mathematical considerations are being used. The problem lies in exactly calculating the multiple inte gral of the type ∞
In =
∫
∞
s 2n – 1
∫
∫
ds 1 ds 2 ⋅⋅⋅
–∞
–∞
2
Trying to follow George Pólya’s advice, let us con sider the following system of ordinary differential equations
⎛ ⎜ d ⎜ ds ⎜ ⎝
2
ds 2n cos ( s 1 – s 2 ) ⋅⋅⋅
–∞ 2
2. “IF YOU CANNOT SOLVE A PROBLEM, THEN THERE COULD BE AN EASIER PROBLEM YOU CAN SOLVE”
(1)
2
× cos ( s 2n – 1 – s 2n ). Because of the s1 ↔ s2 symmetry, the n = 1 case is sim ple ∞
∞
2 2 2 2 1 π I 1 = ds 1 ds 2 ( cos s 1 cos s 2 + sin s 1 sin s 2 ) = , (2) 2! 2
∫
∫
–∞
–∞
since its calculation involves known Fresnel integrals ∞
∫ ds cos s
–∞
∞ 2
=
∫ ds sin s
–∞
2
=
π . 2
⎞ ⎟ ⎟⎛ ⎞ ⎟⎜ x ⎟ ⎟ ⎜ ⎟ . (3) ⎟⎜ y ⎟ ⎟⎝ z ⎠ ⎟ ⎟ ⎠
Such system of equations emerges naturally when one tries to describe a motion of a sphere S2 rolling on the flat surface R2 without slippage. More accurately, it describes the rolling of a sphere of radius R along the Cornu spiral on R2 whose curvature κ = as is propor tional to the arclength s [3, 4]. But what is the relation of this problem to our integral (1)? As the matrices
However, already for n ≥ 2 the above symmetry is lost and things quickly become messy. The n = 2 and n = 3 cases lay at the borderline. They can be done with some efforts even though the calculations become noticeably more involved. Unfortunately, they do not admit an apparent generalization by using the induction method. Already for n = 4 the attempt to use the same methods meets dif ficulties. Nevertheless, we found a way to calculate such type of integrals by invoking some physical arguments. 1 The article is published in the original.
⎛ 2 as ⎜ 0 0 – sin ⎜ 2 ⎞ ⎜ x⎟ 2 1⎜ as y ⎟ = R ⎜ 0 0 cos ⎟ 2 ⎜ z⎠ 2 2 ⎜ as as ⎜ sin – cos 0 2 2 ⎝
⎛ 2 as ⎜ 0 0 – sin ⎜ 2 ⎜ 2 as M(S) = ⎜ 0 0 cos
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