Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 72 / 4 / a / iii

An isolating integral restricts the accessible phase space to a lower-dimensional regular level set. For functionally independent integrals of motion $I_1,\dots ,I_N$, the differentials $d{I}_j$ must be independent, and $I_j=c_j$ impose $N$ independent restrictions. In six-dimensional phase space the common regular level set has dimension $6-N$. Nonisolating orbit labels or multivalued constants do not further reduce the closure explored by the motion. This is the isolating integrals and orbital closure dimension distinction.