By L D Landau and E.M. Lifshitz (Auth.)

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If we denote by no a unit vector in the direction of the velocity of the particle m i after the collision, then the velocities of the two particles after the collision (distinguished by primes) are vio = m^vn^lim^^m^, v^o = -m^vi^l{my^-\-m^. 1) In order to return to the laboratory system, we must a d d to these expressions the velocity V of the centre of mass. no/(mi + = m2) + miV/(mi-hm2). 2) N o further information a b o u t the collision can be obtained from the laws of conservation of momentum and energy.

7) shows t h a t the arbitrary constants a and α are related to Ci and C2 by a = V(cl+c¡\ tan α = -cjc^. 9) Thus, near a position of stable equilibrium, a system excutes har­ monic oscillations. 8) is called the amplitude of the oscillations, and the argument of the cosine is their phase; α is the initial value of the phase, and evidently depends on the choice of the origin of time. The quantity ω is called the angular frequency of the oscillations; in theoretical physics, how­ ever, it is usually called simply the frequency, and we shall use this name henceforward.

Let us consider a case of especial interest, where the external force is itself a simple periodic function of time, of some frequency γ: F(t) =f cos {Yt+β). 2) in the form χ χ = = ¿ e o s ( y / + ^ ) , with the same periodic factor. Substitution in that equation gives b = flm(cú^-y^)\ adding the solution of the homogeneous equation, we obtain the general integral in the form Λ: = α cos ( ω / + α ) + [/Ι^ίρβ-γ^)] cos (yt-\-ß). 4) The arbitrary constants a and α are found from the initial conditions.

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