New PDF release: Classical Mechanics: a graduate course

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Then r = −λ, whose general solution is x = (c1 + c2t) exp(−λt) . If we generalize to systems of n degrees of freedom, the generalized friction forces corresponding to the coordinates xi are linear functions of the velocities . αij xi . fr,i = (29) j Using αik = αki , one can also write fr,i = − ∂F . , ∂ xi . where F = 21 i,j αij xi xj is called the dissipative function. The differential equation is obtained by adding up all these forces to (14) ·· (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 r2 + αij r + kij )Aj = 0 .

28) To seek a diagonal form of {Iij } is equivalent to finding a new system of three axes for which the kinetic energy and the angular momentum take the form given by (27) and (28). In this case the axes are called principal axes of inertia. That means that given an inertial reference system within the body, we can pass from it to the principal axes by a particular orthogonal transformation, which is called transformation to the principal axes. Making equal the components of (22) and (28), we have L1 L2 = Iω1 = I11 ω1 + I12ω2 + I13ω3 = Iω2 = I21 ω1 + I22ω2 + I23ω3 L3 = Iω3 = I31 ω1 + I32 ω2 + I33ω3 .

2 Θ = 2 2 − 1, and by means of equation (44) cot Θ 2Es . = 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. In the nonrelativistic limit, the same result is provided by the quantum mechanical calculations.

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