Classical mechanics by Rosu H.C.

By Rosu H.C.

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R l2 (45) The direction Ψ of the incident asymptote is thus determined by the condition r → ∞: 1 cos Ψ = , that is, according to equation (39), sin Θ 1 = . 2 Then, cot2 Θ = 2 2 − 1, and by means of equation (44) cot 2Es Θ = . 2 ZZ´e2 The functional relationship between the impact parameter and the dispersion angle will be ZZ´e2 Θ s= cot , (46) 2E 2 30 and by effecting the transformation required by the equation (38) we find that σ(Θ) is given by σ(Θ) = 1 4 ZZ´e2 2E 2 csc4 Θ . 2 (47) The equation (47) gives the famous Rutherford scattering cross section derived by him for the dispersion of α particles on atomic nuclei.

Mij xj +kij xj ) = − αij xi . (30) j Employing xk = Ak exp(rt) in (30) and deviding by exp(rt), one can obtain the following system of linear algebraic equations for the constants Aj (mij r 2 + αij r + kij )Aj = 0 . j Making equal to zero the determinant of this system, one gets the corresponding characteristic equation mij r 2 + αij r + kij = 0 . This is an equation for r of degree 2n. 4 NORMAL MODES Before defining the normal modes, we rewrite (15) as follows .. M X + K |X = 0 , where |X is the n-dimensional vector whose matrix representation is (19); M and K are two operators having the matrix representation given by (16) and (17), respectively.

K12 k22 ... k1n ... k2n ..       (17) kn1 kn2 ... knn  ·· (X ) = d2 dt2      x1 x2 ..       (18) xn    (X) =    x1 x2 .. xn 56     .   (19) Similarly to the one dof system, we look for n unknown functions xj (t) of the form xj = Aj exp(iwt) , (20) where Aj are constants to be determined. Substituting (20) in (14) and dividing by exp(iwt), one gets a linear system of algebraic and homogeneous equations, which should be fulfilled by Aj . (−w2mik + kik )Ak = 0 . (21) j This system has nponzero solutions if the determinant of its coefficients is zero.

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