Converting Second-Order ODE to First Order

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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.