Below, we implement the problem of the double pendulum into Maple and solve its dynamics. We start by defining the positions of the masses using the notation in Fig. 1.
The - and -coordinates of the two masses depend on the time, via the time dependence of the angles. Now we can define the kinetic and potential energy as follows:
Thus the Lagrange function is given via (1.2) and (1.3) as:
where in the second step we inserted relations (1.1), used trigonometric relations and simplified via combine and simplify. The equations of motion of the two masses can then be obtained via the Euler-Lagrange equations:
for as:
The resulting equations and govern the dynamics of the two masses. They cannot be solved exactly, but we may try solving them numerically using dsolve. To do so, we set the constants to the following values in SI-units ( is given in , the lengths in and the masses: in )
Furthermore, we set the following initial conditions for the angles and their time derivatives:
with the derivatives given in . Now we can solve the equations of motion (1.7) and (1.9) via
The procedures for the two angles and obtained from this solution:
determine the dynamics of the system. In the following section we explore this solution a bit.