  In this paper we study a problem in oscillations wherein the assumed modes
method offers some analytical and theoretical peculiarities. Specifically, we
study small in-plane oscillations of a slack catenary, or a sagging
inextensible chain fixed at both endpoints. The horizontal and vertical
displacements cannot be approximated independently because of pointwise
inextensibility in the chain. Moreover, the potential energy is a linear
function of the generalized coordinates, and does not directly cause
oscillations. Using assumed modes for the vertical displacements only,
integrating from one endpoint to compute the required horizontal displacements,
treating the horizontal fixity at the distal end as an added scalar constraint,
and obtaining linearized equations, we construct an eigenvalue problem which
contains a Lagrange multiplier. For generic assumed modes, the Lagrange
multiplier is determined by enforcing equilibrium in the undeflected shape.
However, when the modes thus determined are reinserted in the assumed mode
expansion and the calculation done afresh, then the Lagrange multiplier is
indeterminate at first order. Upon retaining terms at the next order, the
distal end fixity constraint introduces quadratic terms into a Lagrangian
without constraints. Results from these two approaches match perfectly. Our
approach offers nontrivial insights into both oscillations and Lagrangian
mechanics. It is also potentially applicable to other problems with
inextensibility in one-dimensional slender members.
