Physics Help

Physics Numericals

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