Dynamics | Part IA, 2003

Write down an expression for the total momentum P\mathbf{P} and angular momentum L\mathbf{L} with respect to an origin OO of a system of nn point particles of masses mim_{i}, position vectors (with respect to O)xiO) \mathbf{x}_{i}, and velocities vi,i=1,,n\mathbf{v}_{i}, i=1, \ldots, n.

Show that with respect to a new origin OO^{\prime} the total momentum P\mathbf{P}^{\prime} and total angular momentum L\mathbf{L}^{\prime} are given by

P=P,L=Lb×P\mathbf{P}^{\prime}=\mathbf{P}, \quad \mathbf{L}^{\prime}=\mathbf{L}-\mathbf{b} \times \mathbf{P}

and hence

LP=LP,\mathbf{L}^{\prime} \cdot \mathbf{P}^{\prime}=\mathbf{L} \cdot \mathbf{P},

where b\mathbf{b} is the constant vector displacement of OO^{\prime} with respect to OO. How does L×P\mathbf{L} \times \mathbf{P} change under change of origin?

Hence show that either

(1) the total momentum vanishes and the total angular momentum is independent of origin, or

(2) by choosing b\mathbf{b} in a way that should be specified, the total angular momentum with respect to OO^{\prime} can be made parallel to the total momentum.

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