By Firk F.W.K.
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15) Integrating this equation leads to ±∫dx/{√(G(x) + D)} = t + D´. 16) Alternatively, if a = d2x/dt2 = g(x) then, multiplying throughout by 2(dx/dt)gives 2(dx/dt)(d 2x/dt2) = 2(dx/dt)g(x). Integrating then gives (dx/dt) 2 = 2∫g(x)dx + D etc. As an example of this method, consider the equation of simple harmonic motion (see later discussion) d2x/dt2 = –ω2x. Multiply throughout by 2(dx/dt), then 2(dx/dt)d2x/dt2 = –2ω2x(dx/dt). This can be integrated to give (dx/dt)2 = –ω2x2 + D. 17) KINEMATICS: THE GEOMETRY OF MOTION 37 (dx/dt)2 = ω2(A2 – x2) = v2, so that dx/dt = ±ω√(A2 – x2).
If x = [x1, x2, x3] is a three-vector that is transformed under x´Tx´ = xT The action of T to give x´ then x = xTx = x 12 + x 22 + x 32 = invariant under . 28) on any three-vector preserves length. The set of all 3×3 orthogonal matrices is denoted by O(3), O(3) = { : T = I, rij ∈ Reals}. The elements of this set satisfy the four group axioms. 6 The relativity of simultaneity: time dilation and length contraction In order to record the time and place of a sequence of events in a particular inertial reference frame, it is necessary to introduce an infinite set of adjacent “observers”, located throughout the entire space.
40) The process of vector addition can be reversed; a vector V can be decomposed into the sum of n vectors of which (n – 1) are arbitrary, and the nth vector closes the polygon. The vectors need not be in the same plane. A special case of this process is the decomposition of a 3-vector into its Cartesian components. A general case V V1 A special case V5 V4 V2 V1, V2, V3, V4 : arbitrary V5 closes the polygon V3 V Vx Vz Vy Vz closes the polygon MATHEMATICAL PRELIMINARIES 24 The vector product of A and B is an axial vector, perpendicular to the plane containing A and B.