Is Noether Symmetric Approach Consistent With Dynamical Equation In Non-minimal Scalar-Tensor Theories?

The form of the coupling of the scalar field with gravity and the potential have been found by applying Noether theorem to two dimensional minisuperspaces in induced gravity model. It has been observed that though the forms thus obtained are consistent with all the equations $\pounds_{X}L=0$, yet they do not satisfy the field equations for $k=\pm 1$, in Robertson-Walker model. It has been pointed out that this is not due to the degeneracy of the Lagrangian, since this problem does not appear in $k=0$ case.It has also been shown that though Noether theorem fails to extract any symmetry from the Lagrangian of such model for $k=\pm 1$, symmetry exists, which can easily be found by studying the continuity equation.