Variational contact symmetries of constrained Lagrangians

Abstract

The investigation of contact symmetries of re-parametrization invariant Lagrangians of finite degrees of freedom and quadratic in the velocities is presented. The main concern of the paper is those symmetry generators which depend linearly in the velocities. A natural extension of the symmetry generator along the lapse function N (t), with the appropriate extension of the dependence inṄ (t) of the gauge function, is assumed; this action yields new results. The central finding is that the integrals of motion are either linear or quadratic in velocities and are generated, respectively by the conformal Killing vector fields and the conformal Killing tensors of the configuration space metric deduced from the kinetic part of the Lagrangian (with appropriate conformal factors). The freedom of re-parametrization allows one to appropriately scale N (t), so that the potential becomes constant; in this case the integrals of motion can be constructed from the Killing fields and Killing tensors of the scaled metric. A rather interesting result is the non-necessity of the gauge function in Noether's theorem due to the presence of the Hamiltonian constraint.

f(T) teleparallel gravity and cosmology

“…We start by constructing the point-like Lagrangian of f (T ) gravity. Starting from the usual action S = d 4 x |e| f (T ) + S m , and following [447][448][449][450][451], one can define a canonical Lagrangian L = L(a,ȧ, T,Ṫ ), with Q = {a, T } the configuration space and T Q = {a,ȧ, T,Ṫ } the related tangent bundle on which L is defined. The scale factor a(t) and the torsion scalar T (t) are taken as independent dynamical variables, and hence one can use the Lagrange mutipliers method in order to set T as a constraint of the dynamics (recall that T = −6H…”

Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities

“…Finally, one can also find higher order generators, i.e. irreducible tensor Killing fields leading to new conserved quantities that do not reduce to the first order ones [28]. An important quantity is the Casimir invariant which commutes with all the Q i 's 16) and is constructed by an element of the universal enveloping algebra spanned by ξ i 's.…”

“…The quantum Casimir operator iŝ 27) which in this case does not coincide with the kinetic part of the Hamiltonian constraint as it happens in most of the cases we study. The corresponding eigenvalue equation iŝ 28) which is Hermitian under the same measure µ as the other operators. The result is…”

Contact symmetries of constrained systems and the associated integrals of motion

“…Symmetries have always played a fundamental role in physical theories. In recent years there exists an increased interest on the subject of Noether point and/or generalized symmetries of physical systems and their geometric nature [1]- [7].…”

“…In general, one can discern two approaches regarding the treatment of symmetries in minisuperspace systems: The first entails an a priori fixation of the gauge (usualy N = 1) at the Lagrangian level and the treatment of the ensuing system as if it were regular ( [8]- [12]). The second method ( [6], [7], [13], [14]) utilizes the gauge invariance of parametrization invariant systems, resulting in the emergence of larger symmetry groups.…”