Converting Second-Order ODE to a First-order System:

Phaser is designed for systems of first-order ordinary differential equations (ODE). Therefore, when faced with a differential equation involving higher-order derivatives, it is necessary to convert it to an equivalent system of first-order equations.

Consider, for instance, the second-order differential equation

    x'' = f(t, x, x')

with the initial conditions

    x(t0) = x0  ,     x'(t0) = x'0   .

To convert this initial-value problem to an equivalent one for a pair of first-order differential equations, introduce the variables

    x1 = x
    x2 = x'   .

Now, the differential equations for x1' and x2' become the following pair

    x1' = x2
    x2' = f(t, x1, x2)

with the initial conditions

    x1 (t0) = x0   ,     x2(t0) = x'0   .

Example: The motions of many simple mechanical systems are governed by Newton's Second Law F = m a , where F is the net force acting on the system, m is the mass, and a is the acceleration which is the second derivative of the displacement with respect to time. Consequently, a = F/m is a natural second-order differential equation.

As a specific example, let us consider the planar pendulum. If we let O be the displacement angle of the pendulum from the vertical position, then, according to Newton's Second Law, the motion of the pendulum is governed by the second-order differential equation

O'' = - (g / l) sin(O) - (c /(l m)) O'   ,

where g is the gravitational constant, l is the length of the pendulum, m is the mass of the pendulum, and c is the friction constant.

To convert this second-order differential equation to an equivalent pair of first-order equations, we introduce the variables

    x1 = O
    x2 = O'  ,

that is, x1 is the angular displacement and x2 is the angular velocity. Now, the equations for x1' and x2' become the following pair

    x1' = x2
    x2' = - (g / l) sin(x1) - (c /(l m)) x2.

These are the equations stored in the ODE Library of Phaser under the name Pendulum ODE.