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I'm trying to workout the poisson bracket [itex]\{L_n, L_m\}[/itex] n,m = 1,2,3.

In the answers, my lecturer has written [itex]\{L_1, L_2\} = \{q_2 p_3 - q_3 p_2 , q_3 p_1 - q_1 p_3\} [/itex] which of course I understand. Then the very next line he writes [itex] = q_2 \{p_3 , q_3\} p_1 + p_2 \{q_3, p_3\} q_1[/itex]

I don't quite see how he has jumped so quickly from one to the other. Has he missed several lines of calculations here, or is he using some property of the Poisson Bracket that I can't see?

Thank you

In the answers, my lecturer has written [itex]\{L_1, L_2\} = \{q_2 p_3 - q_3 p_2 , q_3 p_1 - q_1 p_3\} [/itex] which of course I understand. Then the very next line he writes [itex] = q_2 \{p_3 , q_3\} p_1 + p_2 \{q_3, p_3\} q_1[/itex]

I don't quite see how he has jumped so quickly from one to the other. Has he missed several lines of calculations here, or is he using some property of the Poisson Bracket that I can't see?

Thank you

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