The problem is a nice simple example which can be found in any textbook on Theoetical Mechanics.
The position of the pendulum is described by two generalized coordinates (in spherical polar coordinates) theta and phi (r is constant). Using Lagrange2 equation a system of two second order nonlinear ordinary differential equations arises, which first has to be linearized to a system of 4x now first order ODEs in order to then be solved numerically by one of the matlab buildin solvers.
Here are the lines of code i'm refering to:
y10 = [0.4*pi 0 0 1.5]; % Initial Conditions for [ theta theta' phi phi'] at time t=0
f = @(t,y)[y(2);(( y(4))^2).*sin(y(1)).*cos(y(1))-(g/R).*sin(y(1)) ; y(4);-2.*(cos(y(1)).*y(2).*y(4))./(sin(y(1)))];
[t,y] = ode45(f,tspann,y10); % call ODE45 solver