Variable-mass system events. The following derivation is for a body that is gaining mass (accretion). A body of time-varying mass "m" moves at a velocity v at an initial time "t". In the same instant, a particle of mass dm moves with velocity u. The initial momentum can be written as Now at a time "t" + d"t", let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as Since d"m"dv is the product of two small values, it can be ignored, meaning during d"t"
Angular velocity of bases. For example, changing to the mobile frame: where formula_46 are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles. The angular velocity vector formula_47 defined above may be equivalently expressed as an angular velocity tensor, the matrix (or linear mapping) "W" = "W"("t") defined by: This is an infinitesimal rotation matrix. The linear mapping "W" acts as formula_49: A vector formula_51 undergoing uniform circular motion around an axis satisfies: Given the orientation matrix "A"("t") of a frame, whose columns are the moving orthonormal coordinate vectors