We consider the Lagrangian formulation with duplicated variables of dissipative mechanical systems. The application of Noether theorem leads to physical observable quantities which are not conserved, like energy and angular momentum, and conserved quantities like the Hamiltonian, that generate symmetry transformations and do not correspond to observables. We show that there are simple relations among the equations satisfied by these two types of quantities. In the case of the damped harmonic oscillator, from the quantities obtained by Noether theorem follows the algebra of Feshbach and Tikochinsky. Further, if we consider the whole dynamics, the degrees of freedom separate into a physical and an unphysical sector. We analyze several cases, with linear and nonlinear dissipative forces; the physical consistency of the solutions is ensured observing that the unphysical sector has always the trivial solution.PACS numbers: 04.20.Fy,12.60.Jv * Electronic address: nephtalieliceo@hotmail.com † Electronic address: cramirez@fcfm.buap.mx 2 Hamilton variational principle. In this approach the variation of the action is done with boundary conditions only at the initial time, independently for each of both variables, and at the final time these variables must coincide. A similar development for classical and quantum mechanics was given by means of an extension of the Closed Time Path formalism to classical mechanics by Polonyi [40,41], who in [42] considers the issue of breaking of time reversal symmetry.The main interest in the study of phenomenological dissipative systems is on their quantum description. Classically, the doubled variable formalism allows to write the equations of motion and after that the additional variables are somehow discarded. In fact, these variables are considered as an artifice which takes account of the dissipative external influence, the whole system being isolated. However, from its construction, the nonconservative Lagrangian has not the standard form due to the time reversed characteristics of the additional sector, i.e. the kinetic term is not positive definite and the potential appears with an unstable term. Thus, an interpretation of its outcome as a whole is not obvious. On the other side, in a quantum theory every interacting degree of freedom in general contributes to the probabilities, spectra and mean values, as they form part of the operator algebra. Thus, it would be desirable to consider the classical theory taking into account the whole dynamics. Moreover, a general knowledge of the relevant quantities in the theory, as delivered e.g. by Noether theorem, is necessary for the definition of the Hilbert space. Actually, in the doubled variable approach, Noether theorem has been applied considering the conservation laws of the conservative part, and these laws are violated due to the dissipative terms [41,43]. Furthermore, Noether theorem has been applied in similar approaches to the symmetries of the whole doubled variables action in [44,45], and for time dependent lagrangians in [46]. ...

We apply Noether's theorem to show how the invariances of conservative systems are broken for nonconservative systems, in the variational formulation of Galley. This formulation considers a conservative action, extended by the inclusion of a time reversed sector and a nonconservative generalized potential. We assume that this potential is invariant under the symmetries of the initial conservative system. The breaking occurs because the time reversed sector requires inverse symmetry transformations, under which the nonconservative potential is not invariant. The resulting violation of the conservation laws is consistent with the equations of motion. We generalize this formulation for fermionic and sypersymmetric systems. In the case of a supersymmetric oscillator, the effect of damping is that the bosonic and fermionic components become different frequencies. Considering that initially the nonconservative action is invariant under supersymmetry, and that the breaking is associated to an instability, this result is reminiscent of spontaneous symmetry breaking.

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We study two homogeneous supersymmetric extensions for the f(R) modified gravity model of Starobinsky with the FLRW metric. The actions are defined in terms of a superfield R that contains the FLRW scalar curvature. One model has N = 1 local supersymmetry, and its bosonic sector is the Starobinsky action; the other action has N = 2, its bosonic sector contains, in additional to Starobinsky, a massive scalar field without self-interaction. As expected, the bosonic sectors of these models are consistent with cosmic inflation, as we show by solving numerically the classical dynamics. Inflation is driven by the R2 term during the large curvature regime. In the N = 2 case, the additional scalar field remains in a low energy state during inflation. Further, by means of an additional superfield, we write equivalent tensor-scalar-like actions from which we can give the Hamiltonian formulation.

An extension with local supersymmetry invariance is worked out for the FRW model of Starobinsky. We define an effective curvature superfield R, such that the action depends on R and its supersymmetric covariant derivatives. The bosonic part of the component lagrangian contains R + α 6 R 2 , as well as a massive scalar field. The classical dynamics is solved numerically; it exhibits inflation driven by R 2 , while the scalar field remains in a low energy state. In order to implement the ordinary canonical formulation, we write a classically equivalent action that is not higher-derivative.

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We consider the effective evolution of a phenomenological model from FLRW supersymmetric quantum cosmology with a scalar field. The scalar field acts as a clock and inflaton. We examine a family of simple superpotentials that produce an inflation whose virtual effect on inhomogeneous fluctuations shows very good agreement with PLANCK observational evidence for the tensor-to-scalar ratio and the scalar spectral index.

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