= Solution
Take downward vertical distances from the support. The climber positions are $q_1=L_1$, $q_2=L_1+L_2$ and $q_3=L_1+L_3$. Their <kinetic energy> is therefore
$$
\boxed{T=\frac12m_1\dot L_1^2+\frac12m_2(\dot L_1+\dot L_2)^2+\frac12m_3(\dot L_1+\dot L_3)^2.}
$$
Integrating the tension-extension relation gives the <elastic potential energy> $k(L_i-L_0)^2/2$ for each taut rope. Because downward displacement lowers gravitational potential,
$$
\boxed{U=\frac k2\sum_{i=1}^3(L_i-L_0)^2-g\big[(m_1+m_2+m_3)L_1+m_2L_2+m_3L_3\big],\qquad L=T-U.}
$$
An arbitrary additive constant in $U$ is immaterial. This is the vertical-motion model implicit in using only rope lengths as generalized coordinates; swinging angles are not additional degrees of freedom in this model.
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