这是一份uwa西澳大学PHYS3002的成功案例

Show that the 4-velocity
$$
u^{\mu}=\frac{d x^{\mu}}{d \tau}=\left(\frac{c}{\sqrt{1-v^{2} / c^{2}}}, \frac{\mathbf{v}}{\sqrt{1-v^{2} / c^{2}}}\right)=(c \gamma, \mathbf{v} \gamma)
$$
Show that the condition for such a motion is
$$
w^{\mu} w_{\mu}=\text { constant }=-w_{0}^{2}
$$
where $w_{0}$ is the usual three dimensional acceleration.
Show that in a fixed frame (b) reduces to
$$
\frac{d}{d t} \frac{v}{\sqrt{1-v^{2} / c^{2}}}=w_{0}
$$
Show that
$$
\begin{gathered}
x=\frac{c^{2}}{w_{0}}\left(\sqrt{1+\frac{w_{0}^{2} t^{2}}{c^{2}}}-1\right) \
v=\frac{w_{0} t}{\sqrt{1+w_{0}^{2} t^{2} / c^{2}}}
\end{gathered}
$$

PHYS3002 COURSE NOTES ：

a) Write the Lagrangian and the Hamiltonian, and rewrite them in terms of the variables
$$
\xi=(\varphi+\theta) / \sqrt{2} \quad \eta=(\varphi-\theta) / \sqrt{2}
$$
b) Find an integral of motion other than the energy, and show that its Poisson bracket with $\mathcal{H}$ is zero.
c) Integrate the equations of motion with these initial conditions at $t=$ 0 :
$$
\theta=-\frac{\pi}{4}, \quad \varphi=+\frac{\pi}{4}, \quad \dot{\theta}=\dot{\varphi}=0
$$