= Free-end compatibility for a swimming beam
{title2=$\int_0^L R\,dx=\int_0^L xR\,dx=0$}
For $G_{xx}=R$, zero <bending moment> and transverse force at the leading end give $G(x)=\int_0^x(x-s)R(s)\,ds$. The two trailing-end <free-end bending boundary conditions> hold exactly when the displayed two integrals vanish. They express total transverse force and <torque> balance. Adding unknown rigid translation and yaw to a prescribed deformation supplies the two degrees of freedom of the <swimming recoil correction>. Its principal inertial matrix is the weighted moment matrix of $1$ and $x-x_c$ with weight $m_b+m$, hence is positive definite for a nondegenerate positive mass distribution.
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