# 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(t_{0}) = x_{0} , x'(t_{0}) = x'_{0} .

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

x_{1} = x

x_{2} = x' .

Now, the differential equations for x_{1}' and x_{2}' become the following pair

X1’ = x_{2}

x_{2}' = f(t, x_{1}, x_{2})

With the initial conditions

x_{1} (t_{0}) = x_{0} , x_{2}(t_{0}) = 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

x_{1} = O

x_{2} = O' ,

that is, x_{1} is the angular displacement and x_{2} is the angular velocity. The equations for x_{1}' and x_{2}' become the following pair

x_{1}' = x_{2}

x_{2}' = - (g / l) sin(x_{1}) - (c /(l m)) x_{2}.

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

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