  Hierarchies of Lagrangians of degree two, each only partly determined by the
choice of leading terms and with some coefficients remaining free, are
considered. The free coefficients they contain satisfy the most general
differential geometric criterion currently known for the existence of a
Lagrangian and variational formulation, and derived by solution of the full
inverse problem of the calculus of variations for scalar fourth-order ODEs
respectively. However, our Lagrangians have significantly greater freedom since
our existence conditions are for individual coefficients in the Lagrangian. In
particular, the classes of Lagrangians derived here have four arbitrary or free
functions, including allowing the leading coefficient in the resulting
variational ODEs to be arbitrary, and with models based on the earlier general
criteria for a variational representation being special cases. For different
choices of leading coefficients, the resulting variational equations could also
represent traveling waves of various nonlinear evolution equations, some of
which recover known physical models. Families of regular and embedded solitary
waves are derived for some of these generalized variational ODEs in appropriate
parameter regimes, with the embedded solitons occurring only on isolated curves
in the part of parameter space where they exist. Future work will involve
higher order Lagrangians, the resulting equations of motion, and their solitary
wave solutions.
