Identify 4-Force: Newton's 2nd Law Homework

[PhyPsy]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

 

[mark726]

Thank you for your post. It seems that you have made a few errors in your solution. Firstly, the 4-force should be written as f^a=\gamma(\mathbf{u}\cdot\mathbf{F},\mathbf{F}), not with a negative sign as you have written. Additionally, in your attempt at a solution, you have used the product rule incorrectly. The correct way to apply the product rule in this case would be to have \frac{d(\gamma u_n)}{d\tau}=\frac{d\gamma}{d\tau}u_n+\gamma\frac{du_n}{d\tau}, which would give you the correct terms in your solution.

Furthermore, the \gamma^3 term is not necessary in the final solution. It seems that you have mistakenly multiplied the \gamma term in the 4-momentum by \gamma again in your solution. This is not necessary and should be removed.

Finally, the \gamma^2u_n\mathbf{u}\cdot\mathbf{F} term in your solution should not be present. This term is not present in the book's solution and is a result of your incorrect application of the product rule.

I hope this helps clarify any confusion and helps you arrive at the correct solution. Remember to always double check your calculations and equations to avoid any errors. Keep up the good work!