Generalized Euler Angles for SU(N): Fibration, Invariant Measures, and Explicit Physical Applications

We collect the generalized Euler parametrisation of the unitary groups, in theform developed by Bertini, Cacciatori and Cerchiai and by Tilma and Sudarshan,and use it to carry out six explicit applications with all intermediate stepsdisplayed and every numerical value independently verified. The framework restson the fibration SU(N+1) -> CP^N with fibre U(N), which is what makes theparameters interpretable as generalized Euler angles. After fixing the orthogonal decomposition of the algebra and constructing asection matrix verified to be special unitary to machine precision, we developthe nested construction of the parametrisation for general N, with the partialradii and the telescoping identity that produces the normalised moduli, and thecomplete dictionary relating the angles to the inhomogeneous complexcoordinates. We then fix a single normalisation and derive the volumes of SU(N), U(N) andCP^N from that one choice, checking the fibration identityVol(U(N+1)) = 2 pi Vol(U(N)) Vol(CP^N) at five values of N; the projective volumeis confirmed by a second, independent route. The Berry phase of a closed loop inthe projective coset is computed in closed form and cross-checked againstnumerical integration to machine precision, and the geodesic distance and thekinetic term of the sigma model are obtained explicitly. The space of two-qubitdensity matrices is parametrised and its constraints verified, together with theexact moments of the Schmidt coefficients from the arcsine law. Finally the exactmoments of the Wilson plaquette on SU(2) are derived from the geometry of S^3,which fix the coefficients of the strong-coupling expansion, and the Creutz ratiois evaluated explicitly. Where a standard figure could not be reproduced under the stated normalisation,this is said rather than smoothed over.