Autonomous

A differential equation or system of Ordinary Differential Equations is said to be autonomous if it does not explicitly contain the independent variable (usually denoted $t$). A second-order autonomous differential equation is of the form $F(y,y',y'')=0$, where $y' \equiv {dy/dt} \equiv v$. By the Chain Rule, $y''$ can be expressed as

$$y''=v'={dv \over dt} = {dv \over dy} {dy \over dt} = {dv \over dy} v.$$

For an autonomous ODE, the solution is independent of the time at which the initial conditions are applied. This means that all particles pass through a given point in phase space. A nonautonomous system of $n$ first-order ODEs can be written as an autonomous system of $n+1$ ODEs by letting $t\equiv x_{n+1}$ and increasing the dimension of the system by 1 by adding the equation

$${dx_{n+1} \over dt} = 1.$$