A Study of the Approximate Singular Lagrangian -- Conditional Noether Symmetries and First Integrals

Abstract

The investigation of approximate symmetries of reparametrization invariant Lagrangians of n + 1 degrees of freedom and quadratic velocities is presented. We show that extra conditions emerge which give rise to approximate and conditional Noether symmetries of such constrained actions. The Noether symmetries are the simultaneous conformal Killing vectors of both the kinetic metric and the potential. In order to recover these conditional symmetry generators which would otherwise be lost in gauge fixing the lapse function entering the perturbative Lagrangian, one must consider the lapse among the degrees of freedom. We establish a geometric framework in full generality to determine the admitted Noether symmetries. Additionally, we obtain the corresponding first integrals (modulo a constraint equation). For completeness, we present a pedagogical application of our method.

arXiv commentAccepted to appear in IJGMMP

References in corpus (8)

• Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology
• Lie - point and variational symmetries in minisuperspace Einstein's gravity
• Invariant solutions and Noether symmetries in Hybrid Gravity
• Noether Symmetries as a geometric criterion to select theories of gravity
• Conditional Symmetries and the Canonical Quantization of Constrained Minisuperspace Actions: the Schwarzschild case
• Symmetry, singularities and integrability in complex dynamics III: approximate symmetries and invariants
• Approximate Noether Symmetries and Collineations for Regular Perturbative Lagrangians
• Approximate conditions admitted by classes of the Lagrangian $\mathcal{L}=\frac{1}{2}\left(-u^{\prime 2}+u^2\right)+{\epsilon }^i{G}_i(u,u^{\prime },u^{\prime \prime })$