Ivancevic_Applied-Diff-Geom

68 Applied Differential Geometry: A Modern Introduction(nD) space R n , then when we are given the time and a set of generalizedcoordinates q i we are also given all the points x i of the dynamical system,as the system is determined uniquely. Consequently, the x i are functionsof q i and possibly also of the time, that is,x i = x i (q i , t).If we restrict ourselves to the autonomous dynamical systems in which theseequations do not involve t, i.e.,x i = x i (q i ), (2.18)then differentiating (2.18) with respect to the time t givesẋ i = ∂xi∂q j ˙qj . (2.19)The quantities ˙q i , which form a vector with reference to coordinate transformations(2.15), we shall call the generalized velocity vector. We see from(2.19) that when the generalized velocity vector is given we know the velocityof each point of our system. Further, this gives us the system’s kineticenergy,E kin = 1 2 M αg mn ẋ m α ẋ n α = 1 2 M αg mn∂x m α∂q i∂x n α∂q j ˙qi ˙q j . (2.20)Now, if we use the Euclidean metric tensor g ij to define the materialmetric tensor G ij , including the distribution of all the masses M α of oursystem, asG ij = M α g mn∂x m α∂q i∂x n α∂q j , (2.21)the kinetic energy (2.20) becomes a homogenous quadratic form in the generalizedsystem’s velocities ˙q i ,E kin = 1 2 G ij ˙q i ˙q j . (2.22)From the transformation relation (2.21) we see that the material metrictensor G ij is symmetric in i and j. Also, since E kin is an invariant for alltransformations of generalized coordinates, from (2.22) we conclude thatG ij is a double symmetric tensor. Clearly, this is the central quantityin classical tensor system dynamics. We will see later, that G ij definesthe Riemannian geometry of the system dynamics. For simplicity reasons,