• anomalous precession frequnecy
• beam dynamics
• g-2
• muon anomaly
• phase acceptance

The Muon $g-2$ experiment at Fermilab (E989) aims to measure the anomalous magnetic moment of the muon with an accuracy of 140 ppb (parts per billions). This accuracy, obtained by adding in quadrature a statistical and a systematic contribution of comparable value (100 ppb), will allow to reduce the experimental uncertainty (from the previous E821 experiment at BNL) of a factor of 4, and represents one of the most precise tests of the Standard Model (SM) theory of elementary particles. The first result on the Run-1 dataset was released on April 7, 2021, showing a very good agreement with the previous result from BNL experiment, with a slightly better uncertainty. The corresponding experimental average increases the significance of the discrepancy between the measured and Standard Model prediction of 4.2$\sigma$. The measured quantity is the muon magnetic anomaly a$_\mu$=$\frac{g_\mu-2}{2}$ where g$_\mu$ is the gyromagnetic factor of the muon. Dirac's equation predicts g$_\mu$ = 2, while radiative corrections, dominated by the QED contribution due to an exchange of a virtual photon, causes a per-mille correction on this quantity. By including all the SM contributions, a$_\mu$ is known at 370 ppb. The work of this Thesis focuses on the beam dynamics corrections on $\omega_a$. Due to the prominent role in Run-1 a special attention has been put to the phase-acceptance correction which effect has been evaluated to be within the quoted systematic error on the E821 BNL measurement of the $g-2$.