Longitudinal calculations cheat sheet

Relativistic relationships

Mass energy $E_0=m_0 c^2$
Total energy $E=E_{\mathrm{kin}}+E_0$
Momentum $E=\sqrt{(p c)^2+E_0^2}$ $p=\gamma m_0 \beta c$
Relativistic velocity $\beta = \frac{v}{c} = \frac{p c}{E} = \sqrt{1-\frac{1}{\gamma^2}}$
Lorentz factor $\gamma = \frac{E}{E_0} = \frac{1}{\sqrt{1-\beta^2}}$
Differential relationships $\frac{dp}{p}=\frac{1}{\beta^2}\frac{dE}{E}=\gamma^2\frac{d\beta}{\beta}$

Magnetic rigidity $B\rho = \frac{p}{q}$
Revolution period/frequency $T=\frac{2\pi R}{v}=\frac{1}{f}=\frac{2\pi}{\omega}$
RF period/frequency $f_{\mathrm{rf}}=h f=\frac{\omega_{\mathrm{rf}}}{2\pi}=\frac{1}{T_{\mathrm{rf}}}$
Linear momentum compaction factor (and transition factor) $\alpha_c=\frac{\Delta R/R}{\Delta p/p}=\frac{1}{\gamma^2_{t}}$
Linear phase slippage factor $\eta=\frac{\Delta T/T}{\Delta p/p}=-\frac{\Delta f/f}{\Delta p/p}=\alpha_c-\frac{1}{\gamma^2}$
Other useful relationship $p=\frac{\beta^2 E}{\omega}$

Equation of motion 1 - Phase slippage (drift) along the ring (continuous) $\frac{d\phi}{dt}= \frac{h \eta \omega}{p R} \left(\frac{\Delta E}{\omega}\right)= \frac{h \eta \omega^2}{\beta^2 E} \left(\frac{\Delta E}{\omega}\right)$
Equation of motion 2 - Energy kick with single RF cavity (continuous) $\frac{d}{dt}\left(\frac{E}{\omega}\right)=\frac{qV}{2\pi}\left(\sin\phi-\sin\phi_s\right)$
Equation of motion 1 - Phase slippage (drift) along the ring $\Delta t^{n+1} = \Delta t^{n} + \frac{\eta_{0} T_{\mathrm{rev}}}{\beta_s^2 E_s} \Delta E^{n}$
Equation of motion 2 - Energy kick in cavity $\Delta E^{n+1} = \Delta E^n + q V \sin (\omega_{\mathrm{rf}}\Delta t^{n}) - \left(E_s^{n+1}-E_s^{n}\right)$
Beam energy gain per turn $\left(E_s^{n+1}-E_s^{n}\right)=qV\sin\phi_s$

Bucket height $\Delta E_{\mathrm{max}}=\beta\sqrt{\frac{2qVE}{\pi h |\eta|}}Y\left(\phi_s\right)$
Bucket height reduction factor $Y\left(\phi_s\right)=\left|\cos\phi_s-\frac{\pi-2\phi_s}{2}\sin\phi_s\right|^{1/2}$
Bucket area $A_{\mathrm{bk}}=16\frac{\beta}{\omega_{\mathrm{rf}}}\sqrt{\frac{qVE}{2\pi h |\eta|}}\alpha\left(\phi_s\right)$
Bucket area reduction factor $\alpha\left(\phi_s\right)\approx\frac{1-\sin\phi_s}{1+\sin\phi_s}$
Angular synchrotron frequency (small amplitudes) $\Omega_s^2=2\pi f_s=-\omega^2\frac{h\eta qV\cos\phi_s}{2\pi\beta^2E}$
Non-linear synchrotron frequency for a maximum amplitude in phase $\phi_u$ $\frac{\Omega\left(\phi_u\right)}{\Omega_s}\approx1-\frac{\phi_u^2}{16}$
Synchrotron tune $Q_s = \frac{\Omega_s}{\omega}$

Variables	Equations
$B, p, R$	$\frac{dp}{p}=\gamma_t^2\frac{dR}{R}+\frac{dB}{B}$
$f, p, R$	$\frac{dp}{p}=\gamma^2\frac{df}{f}+\gamma^2\frac{dR}{R}$
$B, f, p$	$\frac{dB}{B}=\gamma_t^2\frac{df}{f}+\frac{\gamma^2-\gamma^2_t}{\gamma^2}\frac{dp}{p}$
$B, f, R$	$\frac{dB}{B}=\gamma^2\frac{df}{f}+\left(\gamma^2-\gamma_t^2\right)\frac{dR}{R}$