# Maximum order of essential and natural conditions

## Main Question or Discussion Point

Is there a physical explanation for why when given a differential equation with spatial derivative of spatial order $2m$ the maximum order of the essential (displacement) boundary condition is $m$, and the maximum order of the natural (force) boundary condition is $2m-1$

for example, the equaiton governing the dynamic behavior of a 1D bar is:
$\frac{\partial ^{2}u}{\partial x^2} = \frac{1}{c^2} \frac{\partial ^{2}u}{\partial t^2}$

the maximum order of the essential boundary condition is 0 :
$u(0,t)=0$
and the maximal order of the natural boundary condition is 1:
$EA\frac{\partial u}{\partial x}(L,t)=R(t)$