Rigid body dynamics The result is a set of m equations of motion that define the dynamics of the rigid body system. If the generalized forces Q are derivable from a potential energy V(q...,q), then these equations of motion take the form In this case, introduce the Lagrangian, L=T-V, so these equations of motion become These are known as Lagrange's equations of motion. The linear and angular momentum of a rigid system of particles is formulated by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles P, i=1...,n be located at the coordinates
Angular velocity origin: The relation between this linear map and the angular velocity pseudovector formula_74 is the following. Because "W" is the derivative of an orthogonal transformation, the bilinear form is skew-symmetric. Thus we can apply the fact of exterior algebra that there is a unique linear form formula_76 on formula_77 that where formula_79 is the exterior product of formula_80 and formula_81. Taking the sharp "L" of "L" we get Introducing formula_83, as the Hodge dual of "L", and applying the definition of the Hodge dual twice supposing that the preferred unit 3-vector is formula_84 where by definition. Because formula_81 is an