Lowering the order by chain rule

When facing a second-order ODE of the form

x^{″} = f (x),

where the independent variable (typically time $t$ ) does not appear explicitly, a method can be used to lower the order.

Setting $v = x^{'}$ (velocity), and using the chain rule,

x^{″} = \frac{d v}{d t} = \frac{d v}{d x} \frac{d x}{d t} = v \frac{d v}{d x},

the second-order ODE becomes a first-order separable ODE in $v$ and $x$ :

v \frac{d v}{d x} = f (x) .

This is sometimes informally called "reducing the order," but it is more properly referred to as lowering the order by velocity substitution or chain rule reduction.

This reduction is closely related to the existence of the Lie symmetry $\partial_{t}$ : invariance under time translations.