The Noether symmetry and the conserved quantity for a fractional Birkhoffian system in terms of Riesz fractional derivatives are studied. The fractional Pfaff variational problems are presented and the fractional Birkhoff’s equations are established within Riesz-Riemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives, respectively. Based on the invariance of the Pfaff action under the infinitesimal transformations, the Noether theorems for the fractional Birkhoffian system are given. The proof of the Noether theorem is done in two steps: first, the Noether theorem under a special one-parameter group of infinitesimal transformations without transforming the time is proved; second, by using a technique of time-reparameterization, the Noether theorem in its general form is obtained. Two examples are given to illustrate the application of the results.

The Liaonan metamorphic core complex (MCC) developed a NNE trend, NWW-dipped, low-angle detachment fault. Ductile shear zone occurred in the lower part and mainly consists of mylonitic granitic gneisses, which indicated a top-to-NWW shearing. Finite strain measurement of feldspar strain markers were estimated by the R_f /φ method. It suggested that strain intensities of mylonites (E_s) changed from 0.36 to 1.0 with an average of 0.75. Fulin and Hossack diagrams indicated strain types were close to flattening strain (lod parameter υ=0.11-0.98). Kinematic vorticity of mylonitic foliations and stretching lineations were estimated by Mohr diagram method and long and short axis method. The results indicated that kinematic vorticity ranged from 0.74 to 0.96 with an average of 0.85, which suggested that the foliations and lineations of mylonites recorded a bulk simple-dominated general shearing. The Kinematic vorticity during formation of extensional crenulation cleavage estimated by C′ method ranged from 0.10 to 0.84 with 0.58 on average. It recorded a later bulk pure-dominated general shearing. Based on the strain measurement, kinematic vorticity and thickness of the detachment fault, the quantification of thinning for the ductile shearing were calculated, which ranged from 130 to 705 m. All results suggested that the deformation mechanism of the Liaonan MCC was represented by earlier simple-dominated shearing and later pure-dominated shearing.