By William B. Heard
This textbook is a latest, concise and targeted remedy of the mathematical options, actual theories and functions of inflexible physique mechanics, bridging the space among the geometric and extra classical ways to the subject. It emphasizes the basics of the topic, stresses the significance of notation, integrates the trendy geometric view of mechanics and gives a wide selection of examples -- starting from molecular dynamics to mechanics of robots and planetary rotational dynamics. the writer has unified his presentation such that utilized mathematicians, mechanical and astro-aerodynamical engineers, physicists, laptop scientists and astronomers can all meet the topic on universal flooring, regardless of their various purposes. * unfastened ideas guide on hand for academics at www.wiley-vch.de/supplements/
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Extra info for Rigid Body Mechanics: Mathematics, Physics and Applications
Line from the north pole of the sphere, say k, through the point r to the place z where the line intersects the complex plane (Fig. 21) Note that points on the equator (ζ = 0) are fixed points of the map, k (ζ = 1) is mapped to the point at infinity, and −k (ζ = −1) is mapped to the origin. The upper hemisphere is mapped outside the unit circle and the lower hemisphere is mapped inside the unit circle. 22) A rotation of R3 induces a single-valued map of the unit sphere into itself and also induces, via the stereographic projection, a map of the complex plane into itself with the same properties.
Then ri = r0 + a + R [(ri − rk ) + (rk − r0 )] = rk + b + R(ri − rk ) where b = r0 − rk + a + R(rk − r0 ) Thus the transformation is again expressed in terms of a translation Tb applied to the new reference point rk and the rotation R which is unchanged by the altered reference point. 4 Rigid Body Transformations Form the Group SE (3) It will now be convenient to take r0 = 0. Then the generic rigid body transformation is r = Rr + a If this is followed by a second rigid body transformation r = Sr + b = SRr + S a + b the result is another rigid body transformation with rotation T = SR and translation c = S a + b.
These circles are mapped to circles in the complex plane (Fig. 8). Note that if the circles are traversed clockwise (seen from the exterior of the sphere) their images will be traversed counterclockwise. 3 Complex Numbers Fig. 8 Rotation of the sphere about n induces, via stereographic projection of the orbits, a map of the complex plane with the fixed point n . the axis of rotation n with the sphere. The details of the argument can be found in the beautifully rendered . We begin by examining the transformation induced by a reflection since any rotation is composed of two reflections.
Rigid Body Mechanics: Mathematics, Physics and Applications by William B. Heard