There is this idea of relativity in Classical Mechanics:
The laws of mechanics valid in an inertial frame must also be valid in any frame moving uniformly with respect to it.
I was just trying to apply these to the case of the law of conservation of momentum and the law of conservation of angular momentum.
Let there be an inertial frame S and another frame S' moving with velocity →v w.r.t to S with:
→r′i=→ri−→vt
→v′i=→vi−→v
For momentum conservation: In frame S', putting ddt∑i→p′i=0 and substituting ddt∑i→pi=0 of frame S in it:
ddt∑i→p′i=ddt∑i→pi−ddt∑imi→v=0−→vddt∑imi
If this has to be 0, then ∑imi=0
Now, on to angular momentum. In frame S:
ddt∑i→Li=ddt∑i(→ri×mi→vi)=0
Am trying to prove the law in frame S' from the law in S:
ddt∑i→L′i=ddt∑i→Li−ddt∑i(→ri×mi→v)−ddt∑i(→vt×mi→vi)
=0−ddt∑i(→ri×mi→v)−ddt∑i(→vt×mi→vi)
=−∑imi(→vi×→v)−∑idmidt(→ri×→v)+∑imi(→vi×→v)−∑imi(→vt×→ai)−∑idmidt(→vt×→vi)
=−∑idmidt(→ri×→v)−∑imi(→vt×→ai)−∑idmidt(→vt×→vi)
=→v×∑idmidt→ri−→vt×∑i→Fi
But this is what I wanted to prove to be 0. I stil have to prove the following:
For a system of particles at →ri with mass mi, which have forces →Fi acting on them such that ∑i→ri×→Fi=0, given ∑idmidt=0; how do I prove:
→v×∑idmidt→ri=→vt×∑i→Fi
for any arbitrary →v and for all time t.
Answer
Using your notation of
r′i=ri−vtv′i=vi−v
and with the assumption that ˙v=0 (uniform motion of frame S') form the linear and angular momentum expressions on the S frame.
p=∑imiviL=∑i(ri×mivi)
Now look at linear and angular momentum in the S' frame and relate them to the ones from S.
p′=∑imiv′i=∑imi(vi−v)=p−(∑imi)v=p−mvL′=∑i(r′i×miv′i)=∑i(ri−vt)×mi(vi−v)=∑i(ri×mivi)−∑i(ri×miv)−vt×(∑imivi)+vt×(∑imi)v=L+v×∑i(miri)+∑i(mivi)×vt
To show that these quantities are conserved, take the derivative (assuming that dpdt=0 and that dLdt=0)
ddtp′=ddtp−mddtv=0ddtL′=ddtL+ddt[v×∑i(miri)]+ddt[∑i(mivi)×vt]=v×∑i(middtri)+ddtp×vt+∑i(mivi)×v=v×∑i(mivi)+p×v=v×p+p×v=0
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