Undergrad First order deviation from circularity

[bengeof]

This is regarding the Bertrand theorem in the book Classical Mechanics by Goldstein. It is said that for more than first order deviations from circularity the orbits are closed only for inverse square law and hooke's law. What does first order deviation mean ?

 

[Delta2]

I think in simple words it means small deviation.

The deviation of a function $f(x)$ from its value $f(x_0)$ can be expressed as taylor series with all the powers of x.

$f(x)-f(x_0)=(x-x_0)f'(x_0)+\frac{1}{2!}(x-x_0)^2f''(x_0)+...+\frac{1}{n!}(x-x_0)^nf^{(n)}(x)+...$

The first term in this series is the first order term(because it involves first power of x and first derivative at $x_0$ ).