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 Lie algebra of the Lie group of rotations of an "n"-dimensional inner product space. In three dimensions, angular velocity can be represented by a pseudovector because second rank tensors are dual to pseudovectors in three dimensions. Since the angular velocity tensor "W" = "W"("t") is a skew-symmetric matrix: its Hodge dual is a vector, which is precisely the previous angular velocity vector formula_60. If we know an initial frame "A"(0) and we are given a "constant" angular velocity tensor "W", we can obtain "A"("t") for any given "t". Recall the matrix differential equation: This equation can be integrated to give: