How was the potential energy of a mass in this mass spring system simplified/approximated? This is from Hobson, Riley, Bence Mathematical Methods, p 322. A spring system is described as follows (they are floating in air like molecules):

enter image description here The equilibrium positions of four equal masses M of a square with sides 2L are $R_n=\pm L_i\pm L_j$ and displacements from equilibrium are $q_n=x_ni+y_nj$.

In the following equation the assumption is made that displacement is much less than equilibrium position $|R_m-T_n|>>|q_m-q_n|$

How was line 2 arrived at from line 1 below? enter image description here


Hint: by the generalised binomial theorem,

$(1+x)^{\frac 12} \approx 1 + \frac 12 x$ for small $x$ (dropping all higher order terms as they're too small to matter).

You are dealing with small $x$ here because of the simplifying assumption that $|R_m - R_n| \gg |q_m - q_n|$, which means the denominator of that term is much larger than the numerator.


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