Topic 2: The pendulum

PHY321F — cp 2005 26
of two coupled first order equations:
dω
dt = −Ω2 sin θ (5)
dθ
dt = ω (6)
where ω is the angular velocity and θ is the angular position.
These equations can now be integrated using the Euler method.
Suppose we know the position and velocity at a time t = i∆t. Then the
above pair of coupled equations can be rewritten as
ω i+1 = ω i − ∆t Ω 2 sin θ i (7)
θ i+1 = θ i + ∆t ω i (8)
(Note that this method can obviously be generalised to solve an arbitrary
system of equations
dy i
dt = f i(y 1 , y 2 , . . . , y N , t)
i = 1, . . . , N
which can be written in a vector notation
dy
dt
= f(y, t)
This is the equation typically solved by a real ODE integrator.)
Write a program (see example) to solve equations (3) and (4), and check that
the result for small initial angular displacement is what you expect.
This solution suffers from a defect: the computed solution does not conserve
the energy E = 1 2 mL2 ω 2 +mgL(1−cos θ). Change your program to calculate
the energy and show this.
A slight modification, however, gives a system which does conserve energy.
ω i+1 = ω i − ∆t Ω 2 sin θ i (9)
θ i+1 = θ i + ∆t ω i+1 (10)
This Euler-Cromer method is a first order area-preserving mapping. Such
mappings exploit the symplectic structure of Hamilton’s equations — they
preserve the canonical structure of these equations.