I was able to prove this, but as far as I can tell this proof holds for all holonomic systems, not just scleronomic ones.
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If a physical system is both a holonomic system and a monogenic system, then it is possible to derive Lagrange's equations from Hamilton's principle.
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This latter is an example of a holonomic system : path integrals in the system depend only upon the initial and final states of the system ( positions in the potential ), completely independent of the trajectory of transition between those states.
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In 1912 ( following earlier incorrect attempts ) Ludwig Schlesinger proved that in general, the deformations which preserve the monodromy data of a ( generic ) Fuchsian system are governed by the integrable holonomic system of partial differential equations which now bear his name:
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The strongest results are obtained for over-determined systems ( holonomic systems ), and on the characteristic variety cut out by the symbols, in the good case for which it is a Lagrangian submanifold of the cotangent bundle of maximal dimension ( involutive systems ).
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