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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