When and why the coffee spills

(a) Either by accelerating too much for a given coffee level (fluid statics)
(b) Or, through more complicated dynamical phenomena:

• Initial acceleration sets an initial sloshing amplitude, which is analogous to the main antisymmetric mode of sloshing.
• This initial perturbation is amplified by the back-and-forth and pitching excitations since their frequency is close to the natural one because of the choice of normal mug dimensions.
• Vertical motion also does not lead to resonance as it is a subharmonic excitation (Faraday phenomena).
• Noise has higher frequency, which makes the antisymmetric mode unstable thus generating a swirl.
• Time to spill depends on "focused"/"unfocused" regime and increases with decreasing maximum acceleration (walking speed).

Lessons learned from sloshing dynamics may suggest strategies to control spilling, e.g. via using

• a flexible container to act as a sloshing absorber in suppressing liquid oscillations.
• a series of concentric rings (baffles) arranged around the inner wall of a container.