Physical laws are independent of choice of standard units.

L

:length

T

:time

M

:mass

O

:temperature

Examples are:

v

=velocity=[L]

accel =

force =

frequency =

Theorem: Physical equations must be homogenous in all units.

general differential equation for , This means that if is the same, the solutions are the same. for gravity.

Buckingham PI theorem.

Transform equation to for unitless (homogenous degree 0). If contain r independent dimensions let . Set . Take the derivative with respect to , then set . You will get r equations with n unknowns. This means there is an which is dimensionless.

General case: where for u s.t. the are dimensionless.

Examples:

• pendulum
• stokes problem
• rain drop
• copper eddy currents

source psfile jl@crush.caltech.edu index