Physics numericals

Even simple physical situations often generate equations for which no known closed-form

solution exists, it is important for physicists to have a toolbox of numerical methods which

can be used to tackle these sorts of problems.

For example, a pendulum suspended by a fixed rod is described by the differential equation

d2

dt2 +

g

l

sin() = 0

where g is the gravitational acceleration and l is length of the rod. The angle the pendulum

makes with the vertical is . This is a classical 2nd order differential equation but it is not linear

since the second term contains a term proportional to sin(). It is a common practise to use the

’small angle approximation’ (sin() for small ) to obtain a solution ((t) = 0 cos(

pg

l t)).

Where 0 is the angle through which we drop the pendulum initially. However if 0 is not small,

the small angle approximation does not work, then we need a (numerical) method of solving

this equation.

The boundary between analytical difficulty and numerical approximation becomes a bit blurred

in a lot of cases and this is where a good knowledge of the problem is useful.

Normally, a numerical method is impervious to minor changes. We simply tweak a parameter

or a line in our code. Such as changing our pendulum equation to remove the small angle

approximation. However analytic methods depend heavily on any changes (e.g. dy

dx+y sin(x) = 0

and dy

dx + y sin(x) = x).

If the problem contains a mis-feature (e.g. discontinuity) a numerical solution may not even

work but an analytic method (where one exists) will always give you a result.

In some cases there is no alternative other than to use a numerical method, for example the

pendulum above, projectile motion when air resistance is taken into account, a simple model

like the Ising model is difficult to solve in 2 dimensions but very easy to write a numerical

simulation. In others the problem is unfeasably large, for example calculating the energy

eigenvalues of a simple molecule can involve calculating eigenvalues of large matrices.

End of Physics numericals