Question 1.8: Prove that the constraints (1.44) are not integrable....

In order to understand the proof that follows, a previous study of Appendix B is necessary. The constraints (1.44) can be expressed as ω^{1} = 0 and ω² = 0 in terms of the 1-forms

ω^{1} =dx − R cos θ d\phi ,  ω² = dy − R sin θ d\phi .  (1.46)

The 2-form Ω defined by Ω = ω^{1} ∧ ω² is given by

Ω = dx ∧ dy − R sin θ dx ∧ d\phi + R cos θ dy ∧ d\phi, (1.47)

and we have

dω^{1} = R sin θ dθ ∧ d\phi , dω² = −R cos θ dθ ∧ d\phi . (1.48)

It follows that

dω^{1} ∧ Ω = R sin θ dx ∧ dy ∧ dθ ∧ d\phi = 0 . (1.49)

According to the Frobenius theorem (Appendix B), the constraints (1.44) are not integrable. Since dω^{1} ∧ Ω = 0 for sin θ = 0, it seems one can only guarantee non-integrability of the constraints (1.44) in the region of the space of variables x, y, θ, \phi such that sin θ ≠ 0. However, an additional computation yields

dω² ∧ Ω = −R cos θ dx ∧ dy ∧ dθ ∧ d\phi , (1.50)

which vanishes only if cos θ = 0. Since cos θ and sin θ do not vanish simultaneously, dω^{1} ∧ Ω and dω² ∧ Ω do not vanish together at any point, with the consequence that the constraints (1.44) are not integrable in the full space of the variables x, y, θ, \phi that describe the configurations of the rolling disc.