Making Equations Dimentionless

It is often useful to remove the units from an equation. We can see which physical mechanisms are more important and the equation to be solved is simpler. We scale all variables so that the variables become dimensionless. We can solve the equation relative to properties of the system e.g. the period.

To make an equation dimensionless one must do the following:

  1. Identify all the independent and dependent variables, e.g.having dimension

  2. Replace each of them with a quantity scaled relative to a characteristic unit e.g. whereis constant and has dimensionandis variable and dimensionless;.

  3. Rewrite the system of equations in terms of their new dimensionless quantities.

  4. Divide through by the coefficient of the highest order derivative term;.

  5. Choose the scale factors so that as many terms as possible have coefficient 1.

Example: Makedimensionless.

1. The independent variable isand the dependent variable is

2. Letandwhereandare constants to be defined later.

3. Sincewe have

4. The coefficient of the highest order term isand dividing by this leads to

5. Chooseand

Example: Makedimensionless whereis the equilibrium position.

This is the equation of the system below.

1. The independent variable isand the dependent variable is

  1. Makeandthen follow the procedure in 3 above to get

Divide through by the leading coefficientto give

5. Chooseand add 1 to both sides to givewhere