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We exhibit a simple motivation of the Dfaz-Higuita-Montesinos formula about the physical degrees of freedom count in terms of Lagrangian parameters. For several singular Lagrangians we illustrate the validity of this fotmula to obtain the number of physical degrees of freedom. Besides, we introduce one arbitrary function by each effective gauge parameter to apply the Lanczos version of the Noether theorem, and thus to show the corresponding genuine constraints and their conservation by the time evolution. [ABSTRACT FROM AUTHOR]

In this paper, we consider certain analytical features of a stochastic model that can explain competition among species and simultaneous predation on the competing species from a geometric perspective. This allows us to build a systematic description of models admitting singular Lagrangians. The model equations are shown to admit a Jacobi Last Multiplier which allows us to construct an appropriate Lagrangian. Due to the singular nature of the Lagrangian, the Hamiltonian formalism may be shown to exist in a submanifold of the cotangent space under certain minimal regularity conditions. In this communication, the Hamiltonian description of the model is constructed via the introduction of Dirac brackets and explicit results for the "Kill the winner" model and its reductions are presented. [ABSTRACT FROM AUTHOR]

In this work, a general theory is given for quantizing singular systems within framework of fractional calculus using the path integral approach. The fractional singular Lagrangians and fractional path integral representation are presented. The fractional Hamilton-Jacobi function is constructed to obtain the path integral quantization for singular systems. One physical example is examined. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the singular Lagrangian systems on the framework of contact geometry. These systems exhibit a dissipative behavior in contrast with the symplectic scenario. We develop a constraint algorithm similar to the presymplectic one studied by Gotay and Nester (the geometrization of the well-known Dirac–Bergmann algorithm). We also construct the Hamiltonian counterpart and prove the equivalence with the Lagrangian side. A Dirac–Jacobi bracket is constructed similar to the Dirac bracket. [ABSTRACT FROM AUTHOR]

Abstract We demonstrate the significance of the Jacobi last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in the activator–inhibitor (AI) systems. We investigate the generalized Hamiltonian dynamics of the AI systems of Turing pattern formation problems, and demonstrate that various subsystems of AI, depending on the choices of parameters, are described either by conformal or contact Hamiltonian dynamics or both. Both these dynamics are subclasses of another dynamics, known as Jacobi mechanics. Furthermore we show that for non Turing pattern formation, like the Gray–Scott model, may actually be described by generalized conformal Hamiltonian dynamics using two Hamiltonians. Finally, we construct a locally defined dissipative Hamiltonian generating function Hudon et al. (2008) of the original system. This generating function coincides with the "free energy" of the associated system if it is a pure conformal class. Examples of pattern formation equation are presented to illustrate the method. [ABSTRACT FROM AUTHOR]

Non-standard Lagrangians have gained recently an increasing interest in the theory of nonlinear differential equations, classical and quantum nonlinear dynamical systems. In this work, we discuss a number of dynamical systems characterized by powers of singular Lagrangians identified to non-standard Lagrangians based on the resulting Hamilton-Jacobi equation. A number of dynamical problems were addressed and a number of statements which support the non-standard Lagrangians formalism were postulated. After connecting the action to a certain complex wave function and in particular for the case of linear potentials, a link is established between the resulting modified Schrödinger equation which describes specific classes of quantum mechanical systems and the Navier-Stokes equation which describes the motion of viscous fluid matters. [ABSTRACT FROM AUTHOR]

In this paper we show how to compute algorithmically the full set of algebraically independent constraints for singular mechanical and field-theoretical models with polynomial Lagrangians. If a model under consideration is not singular as a whole but has domains of dynamical (field) variables where its Lagrangian becomes singular, then our approach allows to detect such domains and compute the relevant constraints. In doing so, we assume that the Lagrangian of a model is a differential polynomial and apply the differential Thomas decomposition algorithm to the Euler–Lagrange equations. [ABSTRACT FROM AUTHOR]

We consider here the analytic classification of pairs ( ω, f) where ω is a germ of a 2-form on the plane and f is a function germ with isolated singularities. We consider the case where ω is singular, i.e., it vanishes nondegenerately along a smooth line H( ω) (Martinet case) and the function f is such that the pair ( f, H( ω)) defines a simple boundary singularity. In analogy with the ordinary case (for symplectic forms on the plane), we show that the moduli in the classification problem are analytic functions of 1-variable and that their number is exactly equal to the Milnor number of the corresponding boundary singularity. Moreover, we derive a normal form for the pair ( ω, f) involving exactly these functional invariants. Finally, we give an application of the results in the theory of constrained Hamiltonian systems, related to the motion of charged particles in the quantization limit in an electromagnetic field, which in turn leads to a list of normal forms of generic singular Lagrangians (of first order in the velocities) on the plane. [ABSTRACT FROM AUTHOR]

Abstract: In the context of metric gravity, we consider a FLRW space–time, filled with a perfect fluid described by a barotropic equation of state ( ). We give the equivalent mini-superspace description and use the reparametrization invariance of the resulting Lagrangian to work in the equivalent constant potential description. At this point, we restrict our analysis to models for which the ensuing scaled mini-superspace is maximally symmetric. As the subsequent analysis shows, such models exhibit the maximum number of autonomous integrals of motion linear in the momenta, which are constructed by the Killing vectors of the respective mini-supermetric. The integrals of motion are used to analytically solve the equations of the corresponding models. A shortcoming of the assumption of maximal symmetry is that it restricts the form of to be ; thus the physical meaning of the ensuing analytical solutions is limited whenever is fixed to a value substantially different from 1. Finally, a brief description of the properties of the resulting Hubble parameters is given. [Copyright &y& Elsevier]

The relationship between Jacobi's last multiplier and the Lagrangian of a second-order ordinary differential equation is quite well known. In this article we demonstrate the significance of the last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in biology. [ABSTRACT FROM AUTHOR]

This paper concerns the Routh reduction procedure for Lagrangians systems with symmetry. It differs from the existing results on geometric Routh reduction in the fact that no regularity conditions on either the Lagrangian L or the momentum map JL are required apart from the momentum being a regular value of JL. The main results of this paper are: the description of a general Routh reduction procedure that preserves the Euler-Lagrange nature of the original system and the presentation of a presymplectic framework for Routh reduced systems. In addition, we provide a detailed description and interpretation of the Euler-Lagrange equations for the reduced system. The proposed procedure includes Lagrangian systems with a non-positively definite kinetic energy metric. [ABSTRACT FROM AUTHOR]

The geometric approach to autonomous classical mechanical systems in terms of a canonical first-order system on the Whitney sum of the tangent and cotangent bundle, developed by Skinner and Rusk, is extended to the time-dependent framework. [Copyright &y& Elsevier]