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Momentum for my exam tomorrow

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1. Homework Statement
Let P be the 4 momentum
u be the 4 velocity
a) Evaluate the Lorentz invariant [itex]P^2[/itex]
b)Differentiate [itex]P^2=P_{\mu}P_{\mu}[/itex] and show that
[tex]\vec{u}\cdot\frac{d}{dt}\left(\frac{m_{0}\vec{u}}{\sqrt{1-u^2/c^2}}\right)=m_{0}c^2\frac{d}{dt}\left(\frac{1}{\sqrt{1-u^2/c^2}}\right)[/tex]

2. The attempt at a solution

The first part yields an answer of [tex]E^2 -p^2 c^2=m^2 c^4[/tex]

Now for part b. Does the P^2 have anything to do with the equality that needs to be proven? Do i need to differentiate P^2 with respect to time? Do i hav to differentiate
[tex]E^2 -p^2 c^2=m^2 c^4[/tex] with respect to time?

Please help!

Thanks
 

Answers and Replies

nrqed
Science Advisor
Homework Helper
Gold Member
3,540
181
1. Homework Statement
Let P be the 4 momentum
u be the 4 velocity
a) Evaluate the Lorentz invariant [itex]P^2[/itex]
b)Differentiate [itex]P^2=P_{\mu}P_{\mu}[/itex] and show that
[tex]\vec{u}\cdot\frac{d}{dt}\left(\frac{m_{0}\vec{u}}{\sqrt{1-u^2/c^2}}\right)=m_{0}c^2\frac{d}{dt}\left(\frac{1}{\sqrt{1-u^2/c^2}}\right)[/tex]

2. The attempt at a solution

The first part yields an answer of [tex]E^2 -p^2 c^2=m^2 c^4[/tex]

Now for part b. Does the P^2 have anything to do with the equality that needs to be proven? Do i need to differentiate P^2 with respect to time? Do i hav to differentiate
[tex]E^2 -p^2 c^2=m^2 c^4[/tex] with respect to time?

Please help!

Thanks
Yes, differentiate with respect to time. The derivative of P^2 obviously gives zero since m^2 c^4 is constant.

Now, write P^2 = E^2 - p dot p c^2

(my small p is the three momentum and dot is the dot product

so d/dt(P^2) = 2 E dE/dt - 2 c^2 p dot dp/dt

This must be zero. Now write E= gamma mc^2 and p = gamma m u where u is the ordinary three velocity
 

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