Nondimensionalizing a physical simulation

Take downward displacement x, constant gravitational acceleration g and linear drag with positive coefficient β. For mass m, the drag relaxation time is τ = m/β. Choose a positive reference length L and a positive reference time T, and write the physical coordinate as L times a dimensionless coordinate X. The first derivative brings one factor of inverse T; the second brings two. Substituting these factors before canceling units avoids the common mistake of rescaling displacement while leaving acceleration unchanged.

Choosing the gravitational time T equal to the square root of L over g makes the dimensionless gravitational acceleration unity. One remaining coefficient measures the ratio of that time to the drag relaxation time. Initial position and velocity must be transformed too. A dimensionless initial velocity of one means a physical speed L/T, rather than one meter per second unless those scales happen to agree. The zero-drag limit is obtained by taking the coefficient to zero, not by dividing through a vanishing drag force.

Changing reference units changes numerical parameter values but preserves dimensionless ratios describing one physical experiment. Changing L as a chosen reference also changes Γ and the scaled observation interval, even for the same physical trajectory; interpreting such changes requires transforming everything consistently. Conversely, two genuinely different physical systems can share a dimensionless initial-value problem and therefore share its scaled solution. Recovering their dimensional trajectories requires each system's own reference scales.

There is no universally best reference time. A weakly damped oscillator often suggests the inverse natural frequency, whereas an overdamped relaxation may be better described by a slower decay time. A poor choice can conceal a rapid transient or leave extremely disparate coefficients. Scaling can expose stiffness but cannot eliminate the physical separation of time scales. For orbital work, resolving a short periapsis passage remains necessary even when the mean period has been scaled to a convenient number.

An absolute position tolerance must be divided by L when solving for X; a relative tolerance remains dimensionless. Velocity tolerances use the velocity scale instead. Combining raw errors in meters and meters per second into an unweighted norm is dimensionally meaningless. Even after scaling, a solver's local tolerance is only an error-control target for its algorithm. Global error, instability, roundoff and inaccuracies in the physical model require separate examination.

A reproducible calculation records the reference scales beside the dimensionless parameters, initial conditions, boundary conditions and output conversion. Otherwise an attractive plot can have an ambiguous time axis. The function lab provides a small scaling check: it evaluates a quadratic and its slope exactly for selected dimensionless inputs. Its controls do not convert units or integrate the falling-body equation, so agreement in that instrument establishes only the arithmetic scaling relationship being tested.