This work deals with chameleon field cosmology (a scalar nonminimally coupled to cold dark matter) in the background of flat Friedmann–Lemaitre–Robertson–Walker (FLRW) space–time. Both classical and quantum have been investigated using Noether symmetry analysis underlying physical system. The Wheeler–DeWitt (WD) equation has constructed on minisuperspace solutions obtained conserved charge.

We search for spherically symmetric solutions of f(R) theories of gravity via the Noether Symmetry Approach. A general formalism in the metric framework is developed considering a point-like f(R) Lagrangian where spherical symmetry is required. Examples of exact solutions are given. PACS numbers: 98.80.-k, 95.35.+x, 95.35.+d, 04.50.+h

Symmetries play important roles in mathematics, physics and mechanics. Since Noether unveiled the profound relations between symmetries and conservation laws, many researches on them were done [1–15]. Recently, symmetry theories have been extended to discrete mechanics and equations [16–25]. As we know, even tiny changes in symmetry, named as perturbation to symmetry, are of great importance fo...