By Jazar R.N.
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Extra info for Advanced Dynamics
Here we present a series solution and employ index notation to provide concise equations. We present the expanded form of the equations in Example 177. 167) in the following form: (t − t0 )2 ... (t − t0 )3 + x i0 +··· 2! 3! 177) s=1 The ε-set, a-set, b-set, and c-set make 84 fundamental parameters that are independent of coordinate systems. Their time derivatives are expressed only by themselves. 168) to develop the series solution of the three-body problem. 3 ORTHOGONAL COORDINATE FRAMES Orthogonal coordinate frames are the most important type of coordinates.
Null element: 4. 51) Example 14 Linear Dependence and Independence The n vectors r1 , r2 , r3 , . . , rn are linearly dependent if there exist n scalars c1 , c2 , c3 , . . 52) The vectors r1 , r2 , r3 , . . , rn are linearly independent if they are not linearly dependent, and it means the n scalars c1 , c2 , c3 , . . 56) and if c2 = 0, we have which shows r1 and r2 are colinear. 57) where at least one of the scalars c1 , c2 , c3 , say c3 , is not zero; then r3 = − c1 c2 r1 − r2 c3 c3 which shows r3 is in the same plane as r1 and r2 .
224) are the parametric equations of the curve. 225) The arc length of a curve is deﬁned as the limit of the diagonal of a rectangular box as the length of the sides uniformly approach zero. 227) where α, β, and γ are the directional cosines of the line. 230) The arc length s is measured from a ﬁxed point on the curve. 247) The circle is made when the parameter θ varies by 2π. The arc length between θ = 0 and θ = π/2 would then be one-fourth the perimeter of the circle. 251) where f (x, y, z) = 0 and g(x, y, z) = 0 represent two surfaces.
Advanced Dynamics by Jazar R.N.