Higher Order Lagrangians Research Articles

Duality in spontaneous broken time translation symmetry: Sisyphus dynamics and the Liénard equation

In this paper, we establish a connection between Sisyphus dynamics and the Liénard-II equation through branched Hamiltonians. Sisyphus dynamics stem from a higher-order Lagrangian. Surprisingly, when expressed in terms of velocity, the Sisyphus dynamical equations align closely with the Liénard-II equation. Sisyphus dynamics introduces velocity-dependent “mass functions”, a departure from conventional position-dependent mass, potentially linked to cosmological time crystals. Additionally, we demonstrate that spontaneously broken time translational symmetry results in a deformed symplectic structure, resembling the classical counterpart of the Generalized Uncertainty Principle (GUP).

Higher-order nonlocal approach to classical and quantum mechanics through non-standard Lagrangians: The case of quantum wells and the quantum states of neutron in earth’s gravitational field

It is well-known that any dynamical system governed by a differential equation containing time derivatives higher than second order unavoidably holds unbounded energy solutions, dubbed ghosts that appear in the Hamiltonian. They correspond to instabilities displayed at the classical level. In this study, we show first that it is possible to construct in classical mechanics, characterized by non-standard Lagrangians and nonlocal-in-time kinetic energy, higher-order derivative theories that avoid the Ostrogradsky ghost. Second, we show that in the realm of quantum mechanics, higher-order discretized energies emerge in the theory which may lead to extended quantum mechanics formalism. The problem of quantum wells has been treated where we showed that negative energy and negative action, complexified energies and complexified actions may emerge in our formalism. We have also discussed the quantum motion of a neutron in the Earth’s gravitational field. It was observed that within the realm of higher-order non-standard Lagrangians, the quantum energies of the neutrons are higher than the energy levels obtained in the basic formalism.

High order conservative Lagrangian scheme for three-temperature radiation hydrodynamics

The three-temperature (3-T) radiation hydrodynamics (RH) equations are widely used in modeling various optically thick high-energy-density-physics environments, such as those in astrophysics and inertial confinement fusion (ICF). In this paper, we will discuss the methodology to construct a high order conservative Lagrangian scheme solving 3-T RH equations. Specifically, the three new energy variables are defined first, in the form of which the three energy equations of the 3-T RH equations are rewritten. The main advantage of this formulation is that it facilitates the design of a scheme with both conservative property and arbitrary high order accuracy. Starting from one dimension and based on the multi-resolution WENO reconstruction and the strong stability preserving (SSP) high order time discretizations, taken as an example, we design a third order conservative Lagrangian scheme both in space and time. To determine the numerical flux for the conservative advection terms in the 3-T RH equations, we propose a HLLC numerical flux which is derived from the divergence theorem rigorously and is suitable for multi-material problems with the ideal-gas equations of state. After that, we discuss how to design a class of high order positivity-preserving explicit Lagrangian schemes to solve the 3-T RH equations which only contain the conservative advection terms in space. Preliminary extension to the two dimensional case is also considered. Finally, various numerical tests are given to verify the desired properties of the high order Lagrangian schemes such as high order accuracy, non-oscillation, conservation and adaptation to multi-material problems.

Action Integral of Regular Systems With Higher Order Lagrangian

In this paper Euler Lagrange equation is studied to obtain the equations of motion of the regular systems with higher order Lagrangian. The solutions of the equations of motion help us to obtain the action integral by substituting the solutions in the given Lagrangian. Then, action integral is typically represented as an integral of the Lagrangian over time, taken along the path of the system between the initial time and the final time of the system. The regular systems with higher order Lagrangian are examined using illustrative example.

Hilbert series and higher-order Lagrangians for the O(N) model

We compare the Hilbert series approach with explicit constructions of higher-order Lagrangians for the O(N) nonlinear sigma model. We use the Hilbert series to find the number and type of operators up to mass dimension 16, for spacetime dimension D up to 12 and N up to 12, and further classify the operators into spacetime parity and parity of the internal symmetry group O(N). The explicit construction of operators is done up to mass dimension 12 for both parities even and dimension 10 for the other three cases. The results of the two methods are in full agreement. This provides evidence for the Hilbert series conjecture regarding co-closed but not co-exact k-forms, which takes into account the integration-by-parts relations.

On a new generalized local fractal derivative operator

In this study, a new generalized local fractal derivative operator is introduced and we discuss its implications in classical systems through the Lagrangian and Hamiltonian formalisms. The variational approach has been proved to be practical to describe dissipative dynamical systems. Besides, the Hamiltonian formalism is characterized by the emergence of auxiliary constraints free from Dirac auxiliary functions. In field theory, it was found that both damped Klein-Gordon and Dirac equations are generalized, and for specific parameters, a field equation comparable to the Barut equation describing the electromagnetic interactions between N spin-1/2 particles in lepton physics is obtained. A Hamiltonian formulation of higher-order Lagrangian has been constructed and discussed as well. The reformulation of the problem based on fractal calculus has been also addressed in details and compared with the basic approach.

Bopp–Podolsky scalar electrodynamics propagators and energy-momentum tensor in covariant and light-front coordinates

We consider the interaction between a charged scalar boson and the Bopp–Podolsky gauge fields. The Bopp–Podolsky (BP) electrodynamics possesses both massive and massless propagation modes for the photon, while preserving gauge invariance. We obtain the propagator of all fields present in the model for the higher-order generalizations of the linear covariant, light-front and doubly transverse light-front gauges. Although BP’s original model is described by a higher-order derivatives Lagrangian, it is possible to work with an equivalent reduced-order version by means of the introduction of an auxiliary vector field. We compute the gauge-invariant improved energy-momentum tensor for the full reduced-order interacting BP model. Besides the more traditional front-form view, we also discuss the light-front perspective in both versions of the model. Within a Lagrangian framework approach, we maintain explicit covariance at all steps and show that the field propagators, as well as the energy-momentum tensor, can be cast into a light-front closed form using specific properties of general coordinate transformations.

Higher-order mesh generation using CISAMR: A case study on bias in presentation and interpretation of results

A modified version of the Conforming to Interface Structured Adaptive Mesh Refinement (CISAMR) algorithm is presented for the construction of higher-order Lagrangian and NURBS-enhanced (NE) finite element meshes. CISAMR non-iteratively transforms a structured grid into a conforming mesh with an upper bound of three on elements aspect ratio. In this work, we introduce new algorithmic aspects for generating higher-order Lagrangian and NE meshes using CISAMR. For each type of element, a comprehensive study is then provided on the performance of first, second, and third-order meshes for solving linear elastic problems with smooth and oscillatory curvilinear edges. Local gradient recovery error and computational cost are used as performance metrics in these examples. Outcomes are then used as a case study to elucidate common sources of bias in presenting and interpreting results. In particular, we show how bias, whether purposeful or not, could lead to misleading conclusions regarding the performance of a method.

Ghost from constraints: a generalization of Ostrogradsky theorem

Ostrogradsky theorem states that Hamiltonian is unbounded when Euler-Lagrange equations are higher than second-order differential equations under the nondegeneracy assumption. Since higher-order nondegenerate Lagrangian can be always recast into an equivalent system with at most first-order derivatives by introducing auxiliary variables and constraints, it is conceivable that the link between ghost and higher derivatives may be reinterpreted as a link between ghost and constraints and/or auxiliary variables. We find that the latter point of view actually provides more general perspective than the former, by exploring the un/boundedness of the Hamiltonian for general theories containing auxiliary variables, for which Euler-Lagrange equations can be essentially second order or lower than that. For Lagrangians including auxiliary variables nonlinearly, we derive the degeneracy condition to evade the Ostrogradsky ghost that can apply even if auxiliary variables can be solved only locally. For theories with constraints with Lagrange multipliers, we establish criteria for inclusion of nonholonomic (velocity-dependent) constraints leading to the absence of local minimum of Hamiltonian. Our criteria include the Ostrogradsky theorem as a special case, and can detect not only ghost associated with higher-order derivatives, but also ghost coming from lower-order derivatives in system with constraints. We discuss how to evade such a ghost. We also provide various specific examples to highlight application and limitation of our general arguments.

High order conservative Lagrangian schemes for one-dimensional radiation hydrodynamics equations in the equilibrium-diffusion limit

Radiation hydrodynamics (RH) describes the interaction between matter and radiation which affects the thermodynamic states and the dynamic flow characteristics of the matter-radiation system. Its application areas are mainly in high-temperature hydrodynamics, including gaseous stars in astrophysics, combustion phenomena, reentry vehicles fusion physics and inertial confinement fusion (ICF). Solving the radiation hydrodynamics equations (RHE), even in the equilibrium-diffusion limit, is a difficult task. In this paper, we will discuss the methodology to construct fully explicit and implicit-explicit (IMEX) high order Lagrangian schemes solving one dimensional RHE in the equilibrium-diffusion limit respectively, which can be used to simulate multi-material problems with the coupling of radiation and hydrodynamics. The schemes are based on the HLLC numerical flux, the essentially non-oscillatory (ENO) reconstruction for the advection term, ENO reconstruction or high order central reconstruction and interpolation for the radiation diffusion term, the Newton iteration method (for the IMEX scheme), and the strong stability preserving (SSP) high order time discretizations. The schemes can maintain conservation and uniformly high order accuracy both in space and time. The issue of positivity-preserving for the high order explicit Lagrangian scheme is also discussed. Various numerical tests for the high order Lagrangian schemes are provided to demonstrate the desired properties of the schemes such as high order accuracy, non-oscillation, and positivity-preserving.

Pseudoinvariance and the extra degree of freedom inf(T)gravity

Nonlinear generalizations of teleparallel gravity entail the modification of a Lagrangian that is pseudoinvariant under local Lorentz transformations of the tetrad field. This procedure consequently leads to the loss of the local pseudoinvariance and the appearance of additional degrees of freedom (d.o.f.). The constraint structure of f(T) gravity suggests the existence of one extra d.o.f. when compared with GR, which should describe some aspect of the orientation of the tetrad. The purpose of this article is to better understand the nature of this extra d.o.f. by means of a toy model that mimics essential features of f(T) gravity. We find that the nonlinear modification of a Lagrangian L possessing a local rotational pseudoinvariance produces two types of solutions. In one case the original gauge-invariant variables -- the analogue of the metric in teleparallelism -- evolve like when governed by the (nondeformed) Lagrangian L; these solutions are characterized by a (selectable) constant value of its Lagrangian, which is the manifestation of the extra d.o.f. In the other case, the solutions do contain new dynamics for the original gauge-invariant variables, but the extra d.o.f. does not materialize because the Lagrangian remains invariant on-shell. Coming back to f(T) gravity, the first case includes solutions where the torsion scalar T is a constant, to be chosen at the initial conditions (extra d.o.f.), and no new dynamics for the metric is expected. The latter case covers those solutions displaying a genuine modified gravity; T is not a constant, but it is (on-shell) invariant under Lorentz transformations depending only on time. Both kinds of f(T) solutions are exemplified in a flat FLRW universe. Finally, we present a toy model for a higher-order Lagrangian with rotational invariance [analogous to f(R) gravity] and derive its constraint structure and number of d.o.f.

Exponential variational integrators for the dynamics of multibody systems with holonomic constraints

We here explore a geometric integrator scheme that is determined by a discretization of a variational principle using a higher-order Lagrangian that uses exponential type of interpolation functions. The resulting exponential variational integrators are here extended to conservative mechanical systems with constraints. To do so we first present continuous Euler-Lagrangian equations with holonomic constraints and then mimic the process for the discrete case. The resulting schemes are then tested to a typical dynamical multibody system with constraints, i.e the double pendulum showing the good long-time behavior when compared to other traditional methods.

Hamiltonian formulation of a class of constrained fourth-order differential equations in the Ostrogradsky framework

We consider a class of Lagrangians that depend not only on some configurational variables and their first time derivatives, but also on second time derivatives, thereby leading to fourth-order evolution equations. The proposed higher-order Lagrangians are obtained by expressing the variables of standard Lagrangians in terms of more basic variables and their time derivatives. The Hamiltonian formulation of the proposed class of models is obtained by means of the Ostrogradsky formalism. The structure of the Hamiltonians for this particular class of models is such that constraints can be introduced in a natural way for eliminating expected instabilities of the fourth-order evolution equations. Moreover, canonical quantization of the constrained equations can be achieved by means of Dirac’s approach to generalized Hamiltonian dynamics.

Higher-order Hamilton dynamics and Hamilton–Jacobi divergence PDE

This paper, using a non-standard Legendrian duality, investigates the Hamiltonian dynamics and formulates a Hamilton–Jacobi type divergence PDE governed by higher-order Lagrangians.

A higher-order Skyrme model

We propose a higher-order Skyrme model with derivative terms of eighth, tenth and twelfth order. Our construction yields simple and easy-to-interpret higher-order Lagrangians. We first show that a Skyrmion with higher-order terms proposed by Marleau has an instability in the form of a baby-Skyrmion string, while the static energies of our construction are positive definite, implying stability against time-independent perturbations. However, we also find that the Hamiltonians of our construction possess two kinds of dynamical instabilities, which may indicate the instability with respect to time-dependent perturbations. Different from the well-known Ostrogradsky instability, the instabilities that we find are intrinsically of nonlinear nature and also due to the fact that even powers of the inverse metric gives a ghost-like higher-order kinetic-like term. The vacuum state is, however, stable. Finally, we show that at sufficiently low energies, our Hamiltonians in the simplest cases, are stable against time-dependent perturbations.

Geometry of special Galileons

A theory known as special Galileon has recently attracted considerable interest due to its peculiar properties. It has been shown that it represents an extremal member of the set of effective field theories with an enhanced soft limit. This property makes its tree-level $S$-matrix fully on-shell reconstructible and representable by means of the Cachazo-He-Yuan representation. The enhanced soft limit is a consequence of new hidden symmetry of the special Galileon action; however, until now, the origin of this peculiar symmetry has remained unclear. In this paper we interpret this symmetry as a special transformation of the coset space $\mathrm{GAL}(D,1)/SO(1,D\ensuremath{-}1)$ and show that there exists a three-parametric family of invariant Galileon actions. The latter family is closed under the duality which appears as a natural generalization of the above mentioned symmetry. We also present a geometric construction of the special Galileon action using a $D$-dimensional brane propagating in $2D$-dimensional flat pseudo-Riemannian space. Within such a framework, the special Galileon symmetry emerges as a $U(1,D\ensuremath{-}1)$ symmetry of the target space, which can be treated as a $D$-dimensional K\"ahler manifold. Such a treatment allows for classification of the higher order invariant Lagrangians needed as counterterms on the quantum level. We also briefly comment on the relation between such higher order Lagrangians and the Lagrangians that are invariant with respect to the polynomial shift symmetry.

New developments in geometric mechanics

We review the concept of a graded bundle, which is a generalisation of a vector bundle, its linearisation, and a double structure of this kind. We then present applications of these structures in geometric mechanics including systems with higher order Lagrangian and the Plateau problem.

Using spectral element method to solve variational inequalities with applications in finance

Under the Black–Scholes model, the value of an American option solves a time dependent variational inequality problem (VIP). In this paper, first we discretize the variational inequality of American option in temporal direction by applying the Rannacher time stepping and achieve a sequence of elliptic variational inequalities. Second we discretize the spatial domain of variational inequalities by using spectral element methods with high order Lagrangian polynomials introduced on Gauss–Legendre–Lobatto points. Also by computing integrals by the Gauss–Legendre–Lobatto quadrature rule we derive a sequence of the linear complementarity problems (LCPs) having a positive definite sparse coefficient matrix. To find the unique solutions of the LCPs, we use the projected successive over-relaxation (PSOR) algorithm. Furthermore we present some existence and uniqueness theorems for the variational inequalities and LCPs. Finally, theoretical results are verified on the relevant numerical examples.

Symplectic structures related with higher order variational problems

In this paper, we derive the symplectic framework for field theories defined by higher order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher order system of partial differential equations to a constrained first-order one, the symplectic structures naturally arising in the dynamics of a first-order Lagrangian theory, and the importance of the Poincaré–Cartan form for variational problems, are all well-established facts. However, their adequate combination corresponding to higher order theories is missing in the literature. Here we obtain a consistent and truly finite-dimensional canonical formalism, as well as a higher order version of the Poincaré–Cartan form. In our exposition, the rigorous global proofs of the main results are always accompanied by their local coordinate descriptions, indispensable to work out practical examples.

Mapping between the classical and pseudoclassical models of a relativistic spinning particle in external bosonic and fermionic fields. II

The exact solution of a system of bilinear identities derived in the first part of our work [1] for the case of real Grassmann-odd tensor aggregate of the type (S,Vμ,Tμν⁎,Aμ,P) is obtained. The consistency of the solution with a corresponding system of bilinear identities including both the tensor variables and their derivatives (S˙,V˙μ,T˙μν⁎,A˙μ,P˙) is considered. The alternative approach in solving of the algebraic system based on introducing complex tensor quantities is discussed. This solution is used in constructing the mapping of the interaction terms of spinning particle with a background (Majorana) fermion field ΨMαi(x). A way of the extension of the obtained results for the case of the Dirac spinors (ψDα,θDα) and a background Dirac field ΨDαi(x), is suggested. It is shown that for the construction of one-to-one correspondence between the most general spinors and the tensor variables, we need a four-fold increase of the number of the tensor ones. A connection with the higher-order derivative Lagrangians for a point particle and in particular, with the Lagrangian suggested by A.M. Polyakov, is proposed.