Given a vector function, I show that , I show that by writing in terms of independent variations in the and direction. I write as a sum of four increments, one purely in the direction, the direction, the direction and the direction as follows:
For this post, I want to prove that in the absence of external forces, the total angular momentum of an N-particle system is conserved.
I start with which is the total momentum of an N-particle system. Now I can vectorially multiple the total momentum by which is the position vector measured from the same origin for each particle. This will give me the total angular momentum of the system which can be written as . After differentiating with respect to , I obtain . In the first cross product, I can substitute with , and since the cross product of any two parallel vectors is zero, the first term becomes zero. This leaves implies that my sum becomes . Now, I can rewrite the net force on particle as , where represents the force exerted on particle by particle . Now I can make a substitution for to give me
…I will finish the rest of this at some point. I seem to have misplaced the book.
Let be defined by the following relations and . I want to show that , where . It must be noted also that the summation is over .
I can start by writing the summation out as . This can be rewritten as due to the fact that which implies , which implies , which implies , and which implies that . Now, I can use another identity, namely that to permute some of the terms so that the expression looks like . Now, I can use two final identities, that to write the expression as , which is equivalent to . After canceling terms, I obtain which is what I wanted to show.