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Identify the 4-force

  • Thread starter PhyPsy
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Homework Statement


Use the 4-dimensional version of Newton's 2nd law to identify the 4-force as [tex]f^a=\gamma(\mathbf{u}\cdot\mathbf{F},\mathbf{F})[/tex], where F is the force acting on the particle.


Homework Equations

[tex]f^a=\frac{dp^a}{d\tau}[/tex][tex]p^a=m_ou^a[/tex][tex]u^a=\gamma(1,\mathbf{u})[/tex]


The Attempt at a Solution


I define the acceleration vector as [tex]\mathbf{a}=(a_1,a_2,a_3)=(\frac{du_1}{d\tau},\frac{du_2}{d\tau},\frac{du_3}{d\tau})[/tex]
and the force vector as [itex]\mathbf{F}=(F_1,F_2,F_3)=(m_oa_1,m_oa_2,m_oa_3)[/itex].
[tex]f^a=m_o(\frac{d\gamma}{d\tau},\frac{d({\gamma}u_1)}{d\tau},\frac{d({\gamma}u_2)}{d\tau},\frac{d({\gamma}u_3)}{d\tau})[/tex][tex]\frac{d\gamma}{d\tau}=-\frac{1}{2}(1-u^2)^{-\frac{3}{2}}\frac{d(1-u^2)}{d\tau}[/tex]I substitute in [itex]u_1^2+u_2^2+u_3^2[/itex] for [itex]u^2[/itex] and [itex]\gamma^3[/itex] for [itex](1-u^2)^{-\frac{3}{2}}[/itex].[tex]\frac{1}{2}\gamma^3(\frac{du_1^2}{d\tau}+\frac{du_2^2}{d\tau}+\frac{du_2^2}{d\tau})=\gamma^3(u_1a_1+u_2a_2+u_3a_3)[/tex]
Here, I use the product rule since both [itex]\gamma[/itex] and [itex]u_n[/itex] are functions of [itex]\tau[/itex].[tex]f^a=m_o(\frac{d\gamma}{d\tau},u_1\frac{d\gamma}{d{\tau}}+\gamma\frac{du_1}{d{\tau}},u_2\frac{d\gamma}{d{\tau}}+\gamma\frac{du_2}{d\tau},u_3\frac{d{\gamma}}{d{\tau}}+\gamma\frac{du_3}{d\tau})[/tex][tex]f^a=m_o(\gamma^3\mathbf{u}\cdot\mathbf{a},\gamma^3u_1\mathbf{u}\cdot\mathbf{a}+{\gamma}a_1,\gamma^3u_2\mathbf{u}\cdot\mathbf{a}+{\gamma}a_2,\gamma^3u_3\mathbf{u}\cdot\mathbf{a}+{\gamma}a_3)[/tex][tex]f^a=\gamma(\gamma^2\mathbf{u}\cdot\mathbf{F},{\gamma}^2u_1\mathbf{u}\cdot\mathbf{F}+F_1,\gamma^2u_2\mathbf{u}\cdot\mathbf{F}+F_2,\gamma^2u_3\mathbf{u}\cdot\mathbf{F}+F_3)[/tex]So I've got 2 things that make my solution different from [itex]f^a=\gamma(\mathbf{u}\cdot\mathbf{F},\mathbf{F})[/itex].
One, I've got this [itex]\gamma^3[/itex] term that is not present in the book's solution.
Two, I've got the [itex]\gamma^2u_n\mathbf{u}\cdot\mathbf{F}[/itex] term in each of the space coordinates that the book's solution does not have.
Does anyone see where I'm messing up?
 

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