By Fowles G.R., Cassiday G.L.
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1 Fixed and Moving Basis The transformation between plane polar coordinates and cartesian coordinates is x = r cos ϕ y = r sin ϕ. 10 For the extension to spherical polar coordinates and to cylindrical coordinates in the three-dimensional case see Appendix C. 42 2 Newtonian Mechanics: First Applications y eϕ er ey ey eϕ r r ϕ er ϕ ex X ex Fig. 6. The ﬁxed (ex , ey ) and moving (er , eϕ ) basis. The trajectory r(t) is indicated by the broken line In addition, it is practical to introduce a ﬁxed (cartesian) and a moving basis (with radial and tangential unit vectors er and eϕ , respectively).
If a closed system has certain invariance properties, then these are generally destroyed by additional external ﬁelds. For example, a homogeneous gravitational ﬁeld destroys the invariance of a free point mass under an (arbitrary) translation in vertical direction. For the potential energy of a point mass in a gravitational ﬁeld one namely has V (x, y, z) = mgz = V (x, y, z + h) = mg (z + h) . In contrast, the translation invariance is conserved in the horizontal directions, V (x, y, z) = mgz = V (x, y + y0 , z) = V (x + x0 , y, z).
1 − (ϕ/ϕ0 )2 With the substitution ϕ = ϕ0 sin β one obtains T =4 ⇒ ω= l g π/2 0 2π = T cos β dβ =4 cos β lπ g2 g . l Who ever looks for exceptional integrals should consult alternatively the integral tables in , see footnote 7. 261 ϕ2 = 4 l g arcsin ϕ ϕ0 ϕ0 =4 0 l π . 2 Motion in a Plane 41 Intermediate Amplitudes One can determine corrections of the harmonic result by expansion of the elliptic integral K in terms of small ϕ0 . With 1 1− =1+ sin2 ϕ20 2 sin α ϕ0 1 sin2 sin2 α + . . 2 2 one obtains K sin ϕ0 2 π/2 = ϕ0 1 sin2 sin2 α + .
Analytical mechanics by Fowles G.R., Cassiday G.L.