Ivancevic_Applied-Diff-Geom

66 Applied Differential Geometry: A Modern IntroductionFig. 2.1 A one–form θ (which is a family of parallel (hyper)surfaces, the so–calledGrassmann planes) pierced by the vector v to give a scalar product θ(v) ≡< θ, v >= 2.6(see text for explanation).is the dimension of the system’s configuration manifold. This viewpoint isthe core of our applied differential geometry.For simplicity, let us suppose that we have a dynamical system withthree DOF (e.g., a particle of mass M, or a rigid body of mass M withone point fixed); generalization to n DOF, with N included masses M α , isstraightforward. The configuration of our system at any time is then givenby three coordinates {q 1 , q 2 , q 3 }. As the coordinates change in value thedynamical system changes its configuration. Obviously, there is an infinitenumber of sets of independent coordinates which will determine the configurationof a dynamical system, but since the position of the system iscompletely given by any one set, these sets of coordinates must be functionallyrelated. Hence, if ¯q i is any other set of coordinates, these quantitiesmust be connected with q i by formulae of the type¯q i = ¯q i (q i ), (i = 1, ..., n(= 3)). (2.15)Relations (2.15) are the equations of transformation from one set of dynamicalcoordinates to another and, in a standard tensorial way (see [Misneret al. (1973)]), we can define tensors relative to this coordinate transformation.The generalized coordinates q i , (i = 1, ..., n) constitute the system’sconfiguration manifold.In particular, in our ordinary Euclidean 3−dimensional (3D) space R 3 ,the ordinary Cartesian axes are x i = {x, y, z}, and the system’s center ofmass (COM) is given byC i =M αx i α∑ Nα=1 M ,αwhere Greek subscript α labels the masses included in the system. If wehave a continuous distribution of matter V = V (M) rather than the dis-