ERRATUM FOR “NONHOLONOMIC AND CONSTRAINED VARIATIONAL MECHANICS”

There is an error in the statement of Theorem 4.25 in [1], a somewhat related typographical error in Remark 4.26, and an error in Remark 4.27 following directly from that in Theorem 4.25. Footnote 8 is also now obsolete. In order to ensure that the errors are unambiguously fixed, what appears below should replace the original text starting from just before the statement of Theorem 4.25 and ending at the end of Section 4.

There is an error in the statement of Theorem 4.25 in [1], a somewhat related typographical error in Remark 4.26, and an error in Remark 4.27 following directly from that in Theorem 4.25. Footnote 8 is also now obsolete. In order to ensure that the errors are unambiguously fixed, what appears below should replace the original text starting from just before the statement of Theorem 4.25 and ending at the end of Section 4.
Next we consider the case of affine vector fields. Here we wish to obtain conditions on a defining subbundle ∆ that ensure that its corresponding affine subbundle variety A(∆) remains in a given subbundle F. However, because A(∆) may be empty, we would like instead to make the problem into one that always has a solution, and then leave the matter of checking whether A(∆) is nonempty to something one can do afterwards. To this end, we note that, if A(∆) ⊆ E is flow-invariant under the affine vector field X aff , then is flow-invariant under X aff , according to Lemma 4.18. Clearly Therefore, we seek conditions on a defining bundle ∆ ⊆ E * ⊕ R M that is flowinvariant under X aff (meaning, by definition, that it is flow-invariant under X aff, * ) and satisfies Λ(∆) Λ(∆) ∩ (E × {1}) ⊆ F. The following result gives conditions to this end, recalling from (10) the definition of ∆ 1 pr 1 (∆).
By letting λ = 0 and g be arbitrary, we see that this implies that a = 1 (of course).
Thus we arrive at the conclusion that condition (i) is equivalent to .
With the preceding observations in place, we can prove the theorem. Finally, we have by part (iii) of the lemma and noting that ∆ 1 = pr 1 (∆). By 3, we have .

ANDREW D. LEWIS
Now let U ⊆ M be open, and let λ ∈ G r Λ(F) (U) and g ∈ C r M (U) and compute Applying pr 1 to this inclusion gives ∇ X0 λ ∈ G r ∆1 (U), which is part (iic) of the theorem.
We conclude, from Proposition 4.13, that, when r = ω or when r = ∞ and F is a subbundle, that all integral curves of X aff with initial conditions in Λ(∆) remain in F. Since Λ(∆) is flow-invariant under X aff (as we pointed out in the preamble to the proof), this implies (i).
One can combine the previous results with Proposition 4.13 to obtain the following procedure for finding invariant affine subbundles contained in a given subbundle.
We first consider the linear case.
Remark 4.26 (Finding invariant cogeneralised subbundles contained in a cogeneralised subbundle). Let r ∈ {∞, ω}, let π : E → M be a C r -vector bundle, let ∇ be a C r -linear connection in E, let F ⊆ E be a C r -cogeneralised subbundle, let X 0 ∈ Γ r (M) be complete, and let A ∈ Γ r (End(E)). Denote Find a flow-invariant cogeneralised subbundle L ⊆ F satisfying the following algebraic/differential conditions: We shall say that L satisfying these conditions is (X lin , F)-admissible. The resulting cogeneralised subbundle L is then flow-invariant under X lin and is contained in F. • In the affine case, we have the following.
Remark 4.27 (Finding invariant affine subbundle varieties contained in a cogeneralised subbundle). Let r ∈ {∞, ω}, let π : E → M be a C r -vector bundle, let ∇ be a C r -linear connection in E, let F ⊆ E be a C r -cogeneralised subbundle, let X 0 ∈ Γ r (M) be complete, let b ∈ Γ r (E), and let A ∈ Γ r (End(E)). Denote Find a flow-invariant defining subbundle ∆ ⊆ E * ⊕ R M satisfying the following algebraic/differential conditions: . We shall say that ∆ satisfying these conditions is (X aff , F)-admissible. Having found such a ∆, check the following: 4. the set S(A(∆)) = {x ∈ M | (0, 1) ∈ ∆ x } is nonempty. The resulting affine subbundle variety A(∆) is then flow-invariant under X aff and is contained in F. • The methodology outlined in the preceding constructions involve some interesting partial differential equations with algebraic constraints. With some effort, it might be possible to apply the integrability theory for partial differential equations [23,24] to arrive at the obstructions to solving these equations. An application of the resulting conditions to the setup of Section 7 would doubtless lead to some interesting answers to the central questions of this paper.