Phase Is a Checked State Ledger

Quarter-Cycle States Fix X, V, and A

A finite phase table pins position, velocity, and acceleration. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

At the right endpoint, acceleration points left

With amplitude 4 metres and angular speed 1 per second, phase zero has position 4 metres, velocity 0, and acceleration negative 4.

x=4 m,v=0,a=−4 m/s2x=4\ \mathrm{m},\quad v=0,\quad a=-4\ \mathrm{m/s}^{2}
Right endpoint stateThe phase check pins x, v, and a.A=4 momega=1 1/sphase=0x=4 mv=0 m/sa=-4 m/s^2

After one quarter cycle, velocity carries the state

At phase 1/4, position is zero, velocity is negative 4 metres per second, and acceleration is zero.

x=0,v=−4 m/s,a=0x=0,\quad v=-4\ \mathrm{m/s},\quad a=0
Quarter-cycle stateThe finite phase table supplies the state.A=4 momega=1 1/sphase=1/4x=0 mv=-4 m/sa=0 m/s^2

Half a cycle flips the position and acceleration signs

At phase 1/2, position is negative 4 metres, velocity is zero, and acceleration is positive 4.

x=−4 m,v=0,a=4 m/s2x=-4\ \mathrm{m},\quad v=0,\quad a=4\ \mathrm{m/s}^{2}
Left endpoint stateAcceleration still points toward the center.A=4 momega=1 1/sphase=1/2x=-4 mv=0 m/sa=4 m/s^2

Three quarters gives the opposite velocity sign

At phase 3/4, position and acceleration are zero while velocity is positive 4 metres per second.

x=0,v=4 m/s,a=0x=0,\quad v=4\ \mathrm{m/s},\quad a=0
Three-quarter-cycle stateThe velocity sign is phase-bound.A=4 momega=1 1/sphase=3/4x=0 mv=4 m/sa=0 m/s^2