## 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. where is constant and has dimension and is 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: Make dimensionless.

1. The independent variable is and the dependent variable is 2. Let and where and are constants to be defined later.

3. Since we have 4. The coefficient of the highest order term is and dividing by this leads to 5. Choose and Example: Make dimensionless where is the equilibrium position.

This is the equation of the system below. 1. The independent variable is and the dependent variable is 1. Make and then follow the procedure in 3 above to get

2. Divide through by the leading coefficient to give  5. Choose and add 1 to both sides to give where  