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27 pages, 457 KB

Open AccessReview

Null Lagrangians and Gauge Functions in Physics: Applications and Recent Developments

by Zdzislaw E. Musielak and Rupam Das

Mathematics 2025, 13(24), 3928; https://doi.org/10.3390/math13243928 - 9 Dec 2025

Cited by 1 | Viewed by 618

The Lagrangian formalism has provided a powerful and elegant framework for obtaining governing equations for classical and quantum systems. It is based on the concept of action, which involves Lagrangians, whose a priori knowledge is required. There are different methods to obtain Lagrangians [...] Read more.

The Lagrangian formalism has provided a powerful and elegant framework for obtaining governing equations for classical and quantum systems. It is based on the concept of action, which involves Lagrangians, whose a priori knowledge is required. There are different methods to obtain Lagrangians for given equations of motion, and a brief review of these methods is presented. However, the main purpose of this review paper is to describe the so-called null Lagrangians and their gauge functions, and discuss their physical applications. The paper also reviews some recent results, which demonstrate that gauge functions play the most fundamental roles in classical dynamics as they can be used to predict the future states of dynamical systems, without solving the equations of motion, as well as to construct their Lagrangians. Full article

(This article belongs to the Special Issue New Developments in Calculus of Variations)

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14 pages, 279 KB

Open AccessArticle

K-Essence Sources of Kerr–Schild Spacetimes

by Bence Juhász and László Árpád Gergely

Universe 2025, 11(3), 100; https://doi.org/10.3390/universe11030100 - 17 Mar 2025

Cited by 1 | Viewed by 1188

Abstract

We extend a result by one of the authors, established for nonvacuum Einstein gravity, to minimally coupled k-essence scalar–tensor theories. First, we prove that in order to source a Kerr–Schild-type spacetime, the k-essence Lagrangian should be at most quadratic in the kinetic term. [...] Read more.

We extend a result by one of the authors, established for nonvacuum Einstein gravity, to minimally coupled k-essence scalar–tensor theories. First, we prove that in order to source a Kerr–Schild-type spacetime, the k-essence Lagrangian should be at most quadratic in the kinetic term. This is reduced to linear dependence when the Kerr–Schild null congruence is autoparallel. Finally, we show that solutions of the Einstein equations linearized in Kerr–Schild-type perturbations are also required to solve the full nonlinear system of Einstein equations, selecting once again k-essence scalar fields with linear Lagrangians in the kinetic term. The only other k-essence sharing the property of sourcing perturbative Kerr–Schild spacetimes, which are also exact, is the scalar field constant along the integral curves of the Kerr–Schild congruence, with the otherwise unrestricted Lagrangian. Full article

(This article belongs to the Section Gravitation)

13 pages, 945 KB

Open AccessArticle

Nonsingular, Lump-like, Scalar Compact Objects in (2 + 1)-Dimensional Einstein Gravity

by Roberto V. Maluf, Gerardo Mora-Pérez, Gonzalo J. Olmo and Diego Rubiera-Garcia

Universe 2024, 10(6), 258; https://doi.org/10.3390/universe10060258 - 11 Jun 2024

Cited by 2 | Viewed by 1553

Abstract

We study the space-time geometry generated by coupling a free scalar field with a noncanonical kinetic term to general relativity in

(2 + 1)

dimensions. After identifying a family of scalar Lagrangians that yield exact analytical solutions in static and circularly [...] Read more.

We study the space-time geometry generated by coupling a free scalar field with a noncanonical kinetic term to general relativity in

(2 + 1)

dimensions. After identifying a family of scalar Lagrangians that yield exact analytical solutions in static and circularly symmetric scenarios, we classify the various types of solutions and focus on a branch that yields asymptotically flat geometries. We show that the solutions within such a branch can be divided in two types, namely naked singularities and nonsingular objects without a center. In the latter, the energy density is localized around a maximum and vanishes only at infinity and at an inner boundary. This boundary has vanishing curvatures and cannot be reached by any time-like or null geodesic in finite affine time. This allows us to consistently interpret such solutions as nonsingular, lump-like, static compact scalar objects whose eventual extension to the

(3 + 1)

-dimensional context could provide structures of astrophysical interest. Full article

(This article belongs to the Collection Open Questions in Black Hole Physics)

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15 pages, 318 KB

Open AccessFeature PaperArticle

Non-Standard and Null Lagrangians for Nonlinear Dynamical Systems and Their Role in Population Dynamics

by Diana T. Pham and Zdzislaw E. Musielak

Mathematics 2023, 11(12), 2671; https://doi.org/10.3390/math11122671 - 12 Jun 2023

Cited by 9 | Viewed by 2328

Abstract

Non-standard Lagrangians do not display any discernible energy-like terms, yet they give the same equations of motion as standard Lagrangians, which have easily identifiable energy-like terms. A new method to derive non-standard Lagrangians for second-order nonlinear differential equations with damping is developed and [...] Read more.

Non-standard Lagrangians do not display any discernible energy-like terms, yet they give the same equations of motion as standard Lagrangians, which have easily identifiable energy-like terms. A new method to derive non-standard Lagrangians for second-order nonlinear differential equations with damping is developed and the limitations of this method are explored. It is shown that the limitations do not exist only for those nonlinear dynamical systems that can be converted into linear ones. The obtained results are applied to selected population dynamics models for which non-standard Lagrangians and their corresponding null Lagrangians and gauge functions are derived, and their roles in the population dynamics are discussed. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

10 pages, 279 KB

Open AccessArticle

The Boundary Homotopy Retract on the Scalar Hairy Charged Black Hole Spacetime

by Mohammed Abu-Saleem and Ali Taani

Axioms 2022, 11(12), 745; https://doi.org/10.3390/axioms11120745 - 19 Dec 2022

Cited by 5 | Viewed by 1937

Abstract

In this paper, we investigate and define the topology of some astrophysical phenomena, like the hairy (scalarized) charged black hole spacetime, to improve our understanding of the kinematics and dynamics of their nature. We use the Lagrangian equation to find different types of [...] Read more.

In this paper, we investigate and define the topology of some astrophysical phenomena, like the hairy (scalarized) charged black hole spacetime, to improve our understanding of the kinematics and dynamics of their nature. We use the Lagrangian equation to find different types of geodesic equations. This can be done under some conditions for the variations of the Cosmological constant and Newton’s constant. We show how to induce the two types (null and spacelike) of geodesics as boundary retractions, in order to obtain the boundary homotopy retract of the scalar charged black hole. These types are used the Lagrangian equation in a 4-D scalar charged black hole to explain the event horizon for this black hole. Full article

(This article belongs to the Special Issue String Theory and Mathematical Physics)

10 pages, 258 KB

Open AccessArticle

Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia

by Zdzislaw E. Musielak

Physics 2021, 3(4), 903-912; https://doi.org/10.3390/physics3040056 - 4 Oct 2021

Cited by 5 | Viewed by 2570

Abstract

New null Lagrangians and gauge functions are derived and they are called nonstandard because their forms are different than those previously found. The invariance of the action is used to make the Lagrangians and gauge functions exact. The first exact nonstandard null Lagrangian [...] Read more.

New null Lagrangians and gauge functions are derived and they are called nonstandard because their forms are different than those previously found. The invariance of the action is used to make the Lagrangians and gauge functions exact. The first exact nonstandard null Lagrangian and its gauge function for the law of inertia are obtained, and their physical implications are discussed. Full article

(This article belongs to the Section Classical Physics)

10 pages, 272 KB

Open AccessArticle

Bateman Oscillators: Caldirola-Kanai and Null Lagrangians and Gauge Functions

by Lesley C. Vestal and Zdzislaw E. Musielak

Physics 2021, 3(2), 449-458; https://doi.org/10.3390/physics3020030 - 12 Jun 2021

Cited by 8 | Viewed by 3328

Abstract

The Lagrange formalism is developed for Bateman oscillators, which includes both damped and amplified systems, and a novel method to derive the Caldirola-Kanai and null Lagrangians is presented. For the null Lagrangians, the corresponding gauge functions are obtained. It is shown that the [...] Read more.

The Lagrange formalism is developed for Bateman oscillators, which includes both damped and amplified systems, and a novel method to derive the Caldirola-Kanai and null Lagrangians is presented. For the null Lagrangians, the corresponding gauge functions are obtained. It is shown that the gauge functions can be used to convert the undriven Bateman oscillators into the driven ones. Applications of the obtained results to quantizatation of the Bateman oscillators are briefly discussed. Full article

(This article belongs to the Section Classical Physics)

20 pages, 1628 KB

Open AccessArticle

Lagrangian Relaxation Based on Improved Proximal Bundle Method for Short-Term Hydrothermal Scheduling

by Zhiyu Yan, Shengli Liao, Chuntian Cheng, Josué Medellín-Azuara and Benxi Liu

Sustainability 2021, 13(9), 4706; https://doi.org/10.3390/su13094706 - 22 Apr 2021

Cited by 5 | Viewed by 2905

Abstract

Short-term hydrothermal scheduling (STHS) can improve water use efficiency, reduce carbon emissions, and increase economic benefits by optimizing the commitment and dispatch of hydro and thermal generating units together. However, limited by the large system scale and complex hydraulic and electrical constraints, STHS [...] Read more.

Short-term hydrothermal scheduling (STHS) can improve water use efficiency, reduce carbon emissions, and increase economic benefits by optimizing the commitment and dispatch of hydro and thermal generating units together. However, limited by the large system scale and complex hydraulic and electrical constraints, STHS poses great challenges in modeling for operators. This paper presents an improved proximal bundle method (IPBM) within the framework of Lagrangian relaxation for STHS, which incorporates the expert system (ES) technique into the proximal bundle method (PBM). In IPBM, initial values of Lagrange multipliers are firstly determined using the linear combination of optimal solutions in the ES. Then, each time PBM declares a null step in the iterations, the solution space is inferred from the ES, and an orthogonal design is performed in the solution space to derive new updates of the Lagrange multipliers. A case study in a large-scale hydrothermal system in China is implemented to demonstrate the effectiveness of the proposed method. Results in different cases indicate that IPBM is superior to standard PBM in global search ability and computational efficiency, providing an alternative for STHS. Full article

(This article belongs to the Special Issue Multi-Utility Energy System Optimization)

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16 pages, 1176 KB

Open AccessArticle

Dynamical Stability and Geometrical Diagnostic of the Power Law K-Essence Dark Energy Model with Interaction

by Bo-Hai Chen, Ya-Bo Wu, Dong-Fang Xu, Wei Dong and Nan Zhang

Universe 2020, 6(12), 244; https://doi.org/10.3390/universe6120244 - 18 Dec 2020

Cited by 4 | Viewed by 2504

Abstract

We investigate the cosmological evolution of the power law k-essence dark energy (DE) model with interaction in FRWL spacetime with the Lagrangian that contains a kinetic function

F (X) = - \sqrt{X} + X

. Concretely, the cosmological evolution in this [...] Read more.

We investigate the cosmological evolution of the power law k-essence dark energy (DE) model with interaction in FRWL spacetime with the Lagrangian that contains a kinetic function

F (X) = - \sqrt{X} + X

. Concretely, the cosmological evolution in this model are discussed by the autonomous dynamical system and its critical points, together with the corresponding cosmological quantities, such as

Ω_{ϕ}

,

w_{ϕ}

,

c_{s}^{2}

, and q, are calculated at each critical point. The evolutionary trajectories are drawn in order to show the dynamical process on the phases plan around the critical points. The result that we obtained indicates that there are four dynamical attractors, and all of them correspond to an accelerating expansion of universe for certain potential parameter and coupling parameter. Besides that, the geometrical diagnostic by the statefinder hierarchy

S_{3}^{(1)}

and

S_{4}^{(1)}

of this scalar field model are numerically obtained by the phase components, as an extended null diagnostic for the cosmological constant. This diagnostic shows that both the potential parameter

λ

and interaction parameter

α

play important roles in the evolution of the statefinder hierarchy. Full article

(This article belongs to the Special Issue Probing the Dark Universe with Theory and Observations)

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11 pages, 263 KB

Open AccessArticle

Gauge Functions in Classical Mechanics: From Undriven to Driven Dynamical Systems

by Zdzislaw E. Musielak, Lesley C. Vestal, Bao D. Tran and Timothy B. Watson

Physics 2020, 2(3), 425-435; https://doi.org/10.3390/physics2030024 - 9 Sep 2020

Cited by 7 | Viewed by 4874

Abstract

Novel gauge functions are introduced to non-relativistic classical mechanics and used to define forces. The obtained results show that the gauge functions directly affect the energy function and allow for converting an undriven physical system into a driven one. This is a novel [...] Read more.

Novel gauge functions are introduced to non-relativistic classical mechanics and used to define forces. The obtained results show that the gauge functions directly affect the energy function and allow for converting an undriven physical system into a driven one. This is a novel phenomenon in dynamics that resembles the role of gauges in quantum field theories. Full article

(This article belongs to the Section Classical Physics)

17 pages, 296 KB

Open AccessArticle

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

by Zdzislaw E. Musielak, Niyousha Davachi and Marialis Rosario-Franco

Mathematics 2020, 8(3), 379; https://doi.org/10.3390/math8030379 - 9 Mar 2020

Cited by 22 | Viewed by 5093

Abstract

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation [...] Read more.

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed. Full article

(This article belongs to the Special Issue Special Functions and Applications)

33 pages, 343 KB

Open AccessReview

Distribution Function of the Atoms of Spacetime and the Nature of Gravity

by Thanu Padmanabhan

Entropy 2015, 17(11), 7420-7452; https://doi.org/10.3390/e17117420 - 28 Oct 2015

Cited by 37 | Viewed by 5428

Abstract

The fact that the equations of motion for matter remain invariant when a constant is added to the Lagrangian suggests postulating that the field equations of gravity should also respect this symmetry. This principle implies that: (1) the metric cannot be varied in [...] Read more.

The fact that the equations of motion for matter remain invariant when a constant is added to the Lagrangian suggests postulating that the field equations of gravity should also respect this symmetry. This principle implies that: (1) the metric cannot be varied in any extremum principle to obtain the field equations; and (2) the stress-tensor of matter should appear in the variational principle through the combination T_abn^an^bwhere n_a is an auxiliary null vector field, which could be varied to get the field equations. This procedure uniquely selects the Lanczos–Lovelock models of gravity in D-dimensions and Einstein’s theory in D = 4. Identifying na with the normals to the null surfaces in the spacetime in the macroscopic limit leads to a thermodynamic interpretation for gravity. Several geometrical variables and the equation describing the spacetime evolution acquire a thermodynamic interpretation. Extending these ideas one level deeper, we can obtain this variational principle from a distribution function for the “atoms of spacetime”, which counts the number of microscopic degrees of freedom of the geometry. This is based on the curious fact that the renormalized spacetime endows each event with zero volume, but finite area! Full article

(This article belongs to the Special Issue Entropy in Quantum Gravity and Quantum Cosmology)

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