Reduction Higher Order Equations Systems


Phaser is designed for systems of first-order ordinary differential equations (ODE). Therefore, when faced with a differential equation involving higher-order derivative, 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, which 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 mass, a is acceleration which is the second derivative of the displacement with respect to time. a = F/m is a natural second-order differential equation.

As a specific example, let us consider the planar pendulum. In case if we let O be the displacement angle of the pendulum from the vertical position, according to Newton's Second Law, 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 mass of the pendulum, c is friction constant.

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

    x1 = O
    x2 = O'  ,

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

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

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

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