Where the position and velocity the current time step are updated based the values position

The Taylor expansion formulas for positions and velocities with truncation error O(dt^4) and O(dt^2), respectively, can be written as follows:

Position Update Formula:

• x(t) represents the position at time t.

• v(t) represents the velocity at time t.

These formulas can be used to numerically integrate the equations of motion for a system, where the position and velocity at the current time step are updated based on the values of position, velocity, acceleration, and jerk at that time step.