Degenerate Lagrangians Research Articles

Generalized fractional bi-Hamiltonian structure of Plebański’s second heavenly equation in terms of conformable fractional derivatives

In this paper, we propose a symplectic and bi-Hamiltonian structure to a generalized (3+1) -dimensional Plebański’s second heavenly (SH) equation by using a conformable fractional derivative. We write the SH equation in terms of conformable fractional derivatives by replacing classical time (t) and space (z) derivatives with conformable fractional derivatives. We name this equation as a conformable fractional second heavenly (CFSH) equation and cast it in a two-component evolutionary form. We construct a degenerate Lagrangian for a two-component conformable fractional system and find a new Euler–Lagrange formalism for four-dimensional Lagrange density. We apply Dirac’s theory of constraints to the CFSH equation and we prove that it admits bi-Hamiltonian structure according to Magri’s theorem.

Symmetry and reduction for second order degenerate Lagrangians

For the chiral oscillator described by a second order and degenerate Lagrangian with special Euclidean group of symmetries, we show, by cotangent bundle Hamiltonian reduction, that reduced equations are Lie-Poisson on dual of oscillator algebra, the central extension of special Euclidean algebra in two dimensions.

Improved accuracy in degenerate variational integrators for guiding centre and magnetic field line flow

First-order-accurate degenerate variational integration (DVI) was introduced in Ellison et al. (Phys. Plasmas, vol. 25, 2018, 052502) for systems with a degenerate Lagrangian, i.e. one in which the velocity-space Hessian is singular. In this paper we introduce second-order-accurate DVI schemes, both with and without non-uniform time stepping. We show that it is not in general possible to construct a second-order scheme with a preserved two-form by composing a first-order scheme with its adjoint, and discuss the conditions under which such a composition is possible. We build two classes of second-order-accurate DVI schemes. We test these second-order schemes numerically on two systems having non-canonical variables, namely the magnetic field line and guiding centre systems. Variational integration for Hamiltonian systems with non-uniform time steps, in terms of an extended phase space Hamiltonian, is generalized to non-canonical variables. It is shown that preservation of proper degeneracy leads to single-step (one-step) methods without parasitic modes, i.e. to non-uniform time step DVIs. This extension applies to second-order-accurate as well as first-order schemes, and can be applied to adapt the time stepping to an error estimate.

Dirac’s analysis and Ostrogradskii’s theorem for a class of second-order degenerate Lagrangians

This paper analyzes the constraint structure of a class of degenerate second-order particle Lagrangian that includes chiral oscillator, noncommutative oscillator, and two examples from reduced topologically massive gravity. For even-dimensional configuration spaces with maximal nondegeneracy, Dirac bracket is defined solely by coefficient field of highest derivative whereas for odd dimensions almost all fields may contribute. Ostrogradskii’s theorem on energy instability is discussed. Results of Dirac analysis are used to identify ghost degrees of freedom. Translational symmetries are used to construct first-order variational formalisms for oscillator examples, thereby making them ghost-free.

Slow manifolds of classical Pauli particle enable structure-preserving geometric algorithms for guiding center dynamics

Since variational symplectic integrators for the guiding center was proposed [1,2], structure-preserving geometric algorithms have become an active research field in plasma physics. We found that the slow manifolds of the classical Pauli particle enable a family of structure-preserving geometric algorithms for guiding center dynamics with long-term stability and accuracy. This discovery overcomes the difficulty associated with the unstable parasitic modes for variational symplectic integrators when applied to the degenerate guiding center Lagrangian. It is a pleasant surprise that Pauli's Hamiltonian for electrons, which predated the Dirac equation and marks the beginning of particle physics, reappears in classical physics as an effective algorithm for solving an important plasma physics problem. This technique is applicable to other degenerate Lagrangians reduced from regular Lagrangians.

Evolutionary equations and constraints: Maxwell equations

By fixing a reference frame in spacetime, it is possible to split the Euler-Lagrange equations associated with a degenerate Lagrangian into purely evolutionary equations and constraints on the allowed Cauchy data with respect to the notion of space and time associated with the given reference frame. In the context of classical electrodynamics, we introduce a “covariantization procedure” that allows us to invert the perspective and to recover a full-fledged covariant formulation of Maxwell’s equations starting only with constraint equations (i.e., Gauss law and the local version of the law of conservation of magnetic flux) as perceived in a sufficient number of (inertial) reference frames on Minkowski spacetime.

Modified equations for variational integrators applied to Lagrangians linear in velocities

Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.

Reductions of topologically massive gravity I: Hamiltonian analysis of second order degenerate Lagrangians

We present Skinner-Rusk and Hamiltonian formalisms of second order degenerate Clément and Sarıoğlu-Tekin Lagrangians. The Dirac-Bergmann constraint algorithm is employed to obtain Hamiltonian realizations of Lagrangian theories. The Gotay-Nester-Hinds algorithm is used to investigate Skinner-Rusk formalisms of these systems.

Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order

We present all scalar-tensor Lagrangians that are cubic in second derivatives of a scalar field, and that are degenerate, hence avoiding Ostrogradsky instabilities. Thanks to the existence of constraints, they propagate no more than three degrees of freedom, despite having higher order equations of motion. We also determine the viable combinations of previously identified quadratic degenerate Lagrangians and the newly established cubic ones. Finally, we study whether the new theories are connected to known scalar-tensor theories such as Horndeski and beyond Horndeski, through conformal or disformal transformations.

Hamiltonian analysis of higher derivative scalar-tensor theories

We perform a Hamiltonian analysis of a large class of scalar-tensor Lagrangians which depend quadratically on the second derivatives of a scalar field. By resorting to a convenient choice of dynamical variables, we show that the Hamiltonian can be written in a very simple form, where the Hamiltonian and the momentum constraints are easily identified. In the case of degenerate Lagrangians, which include the Horndeski and beyond Horndeski quartic Lagrangians, our analysis confirms that the dimension of the physical phase space is reduced by the primary and secondary constraints due to the degeneracy, thus leading to the elimination of the dangerous Ostrogradsky ghost. We also present the Hamiltonian formulation for nondegenerate theories and find that they contain four degrees of freedom, including a ghost, as expected. We finally discuss the status of the unitary gauge from the Hamiltonian perspective.

Fractional variational approach with non-standard power-law degenerate Lagrangians and a generalized derivative operator

We extend the fractional actionlike variational approach where we substitute the standard Lagrangian by a non-standard power-law Lagrangian holding a generalized derivative operator. We focus on degenerate Lagrangians for the constructed fractional formalism where we show that non-linear oscillators with damping solutions may be obtained from degenerate non-standard Lagrangians which are linear in velocities. We explore as well the case of $2^{nd}$-order derivatives non-standard Lagrangians and we study the case where Lagrangians are linear in accelerations where damping solutions are obtained as well. It was observed that these extensions give another possibility to obtain more fundamental aspects which may have interesting classical effects.

Modeling chemical reaction networks on the Pontryagin bundle with the Hamilton-Pontryagin approach

The Lagrange-d'Alembert-Pontryagin principle is a versatile approach to model dynamical systems including resistive forces from the Lagrangian view. We show in this work, how this method can be applied to open stoichiometric reaction networks. We define a degenerate Lagrangian as the sum of the chemical potential of the different chemical species, and define the Lagrangian forces as connection between chemical affinities and the reaction fluxes. We used this formulation to apply a variational approach including only one kinetic parameter per chemical reaction. Moreover, the stoichiometry of the network is used to define a Dirac structure, which includes all stoichiometric constraints and is defined as the Whitney sum of a distribution and its annihilator. By these means we derive the equations for mass-action kinetics as implicit Euler-Lagrange equations. This results in possible access to variational optimization methods and a direct access to variational integrators for biochemical reaction networks.

Higher-Stage Noether Identities and Second Noether Theorems

The direct and inverse second Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Such Lagrangian theory is characterized by a hierarchy of nontrivial higher-stage Noether identities which is described in the homology terms. If a certain homology regularity condition holds, one can associate with a reducible degenerate Lagrangian the exact Koszul–Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete nontrivial Noether and higher-stage Noether identities. The second Noether theorems associate with the above-mentioned Koszul–Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this operator is extended to the nilpotent BRST operator which brings the above-mentioned cochain sequence into the BRST complex and provides a BRST extension of an original Lagrangian.

Formulation of singular theories in a partial Hamiltonian formalism using a new bracket and multi-time dynamics

A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The noncanonical generalized velocities satisfy a system of linear algebraic equations, which allows an appropriate classification of singular theories (gauge and nongauge). A new antisymmetric bracket (similar to the Poisson bracket) is introduced, which describes the time evolution of physical quantities in a singular theory. The origin of constraints is shown to be a consequence of the (unneeded in our formulation) extension of the phase space, when the new bracket transforms into the Dirac bracket. Quantization is briefly discussed.

A Generalized Nonlinear Oscillator From Non-Standard Degenerate Lagrangians and Its Consequent Hamiltonian Formalism

Nonlinear oscillators play an important role in science and engineering and they exist in many types depending on the physical problem and its corresponding dynamical system. Two important classes of nonlinear oscillators are (1) those characterized by the unusual property “a frequency completely independent of the amplitude and is the same as that of the linear harmonic oscillator” due to its significant impacts in nonlinear dynamics, chaos theory and pattern formation and (2) those characterized by a “position-dependent mass” as introduced by Mathews and Lakshmanan due to their relevance in describing many fundamental properties in science like quantum optics and metal clusters. These oscillators are also described by non-standard Lagrangians which is an important class of functions existing in some group of dissipative dynamical systems. Deriving the equations of motion for systems holding non-standard Lagrangians is a very motivating open problem and requires particular awareness. Motivated by these features, the main goal of this work is to show that a comparable nonlinear differential equation of a non-linear oscillator with periodic solutions to the one studied by Mathew and Lakshmanan may be derived as well from non-standard degenerate complexified Lagrangians. More precisely, we will prove that the equation of motion for such types of nonlinear oscillators with periodic solutions may be derived as well from non-standard degenerate Lagrangians which are linear in velocities. The generalized Hamiltonian formulation for the case of non-standard degenerate Lagrangians and based on Dirac’s theory for degenerate systems was also discussed.

Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints

We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton–Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac–Hamilton–Jacobi equation reduces to a variant of the nonholonomic Hamilton–Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac–Hamilton–Jacobi equation can be used to exactly integrate the equations of motion.

On dark energy models of single scalar field

In this paper we revisit the dynamical dark energy model building based on single scalar field involving higher derivative terms. By imposing a degenerate condition on the higher derivatives in curved spacetime, one can select the models which are free from the ghost mode and the equation of state is able to cross the cosmological constant boundary smoothly, dynamically violate the null energy condition. Generally the Lagrangian of this type of dark energy models depends on the second derivatives linearly. It behaves like an imperfect fluid, thus its cosmological perturbation theory needs to be generalized. We also study such a model with explicit form of degenerate Lagrangian and show that its equation of state may cross -1 without any instability.

Dirac cotangent bundle reduction

The authors' recent paper in Reports in Mathematical Physics develops Dirac reduction for cotangent bundles of Lie groups, which is called Lie--Dirac reduction . This procedure simultaneously includes Lagrangian, Hamiltonian, and a variational view of reduction. The goal of the present paper is to generalize Lie--Dirac reduction to the case of a general configuration manifold; we refer to this as Dirac cotangent bundle reduction . This reduction procedure encompasses, in particular, a reduction theory for Hamiltonian as well as implicit Lagrangian systems, including the case of degenerate Lagrangians.      First of all, we establish a reduction theory starting with the Hamilton-Pontryagin variational principle, which enables one to formulate an implicit analogue of the Lagrange-Poincaré equations. To do this, we assume that a Lie group acts freely and properly on a configuration manifold, in which case there is an associated principal bundle and we choose a principal connection. Then, we develop a reduction theory for the canonical Dirac structure on the cotangent bundle to induce a gauged Dirac structure . Second, it is shown that by making use of the gauged Dirac structure, oneobtains a reduction procedure for standard implicit Lagrangian systems, whichis called Lagrange-Poincaré-Dirac reduction . This procedurenaturally induces the horizontal and vertical implicitLagrange-Poincaré equations , which are consistent with thosederived from the reduced Hamilton-Pontryagin principle. Further, we develop the case in which a Hamiltonian is given (perhaps, but notnecessarily, coming from a regular Lagrangian); namely, Hamilton-Poincaré-Dirac reduction for the horizontal andvertical Hamilton-Poincaré equations . We illustrate the reductionprocedures by an example of a satellite with a rotor.      The present work is done in a way that is consistent with, and may be viewed as a specialization of the larger context of Dirac reduction, which allows for Dirac reduction by stages . This is explored in a paper in preparation by Cendra, Marsden, Ratiu and Yoshimura.

Dirac constraint analysis and symplectic structure of anti-self-dual Yang-Mills equations

We present the explicit form of the symplectic structure of anti-self-dual Yang-Mills (ASDYM) equations in Yang’s J- and K-gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac’s theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang-Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both J- and K-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten-Zuckerman formalism. We show that the appropriate component of the Witten-Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac’s theory. Finally, we present the Backlund transformation between the J- and K-gauges in order to apply Magri’s theorem to the respective two Hamiltonian structures.

Inequivalent Lagrangians for the damped harmonic oscillator

A constant of the motion, in addition to what exists in the literature, is presented for the damped harmonic oscillator and its dynamical origin is investigated. These two constants of motion are used to construct expressions for a hierarchy of inequivalent Lagrangians. It is shown that each inequivalent Lagrangian may be related to a higher order degenerate Lagrangian. The hierarchical Lagrangians tend to pose some characteristic problems for discussing the corresponding phase-space structure. PACS Nos.: 47.20.Ky, 42.81.Dp