**Original manuscript:** 2020/08/30

Restricting ourselves to kinetic minus potential energy Lagrangians, we are
concerned with answering the following questions: when is a nonholonomic
trajectory considered a constrained variational trajectory? When are some
nonholonomic trajectories considered constrained variational trajectories?
And finally when are all nonholonomic trajectories considered constrained
variational trajectories? In Lewis [Nonholonomic and variational mechanics,
*Journal of Geometric Mechanics*, **12**(2), 165-308, 2020], these
questions are given answers in terms of conditions on the existence of
flow-invariant affine subbundle varieties and cogeneralized subbundles
contained in a cogeneralized subbundles, depending on whether the
trajectories are regular or singular, respectively. These answers were shown
to be complete in the analytic case but required the assumption of a locally
constant rank in the smooth case. Moreover, differential and algebraic
conditions were found for this inclusion problem. Carrying on from these
results, we give a geometric formulation of a partial differential equation
representing the differential condition and prove its formal integrability
using Spencer theory. Then we investigate how to rewrite the above
conditions if we have local solutions of these PDEs to answer the above
questions.

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Last Updated: Fri Mar 15 08:14:55 2024