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We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on [Formula: see text], which in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method, we obtain the first integral of the SORE and the results are applied to the study of its Lagrangian and Hamiltonian descriptions. Using these results, we show the existence of a Lagrangian description for SORE, and the Painlevé II equation is analyzed.

A noncentral force is a prototypical example of nonconservative position dependent force, known as curl force; this terminology was first proposed by Berry and Shukla in [1]. In this paper, we extend the construction of a noncentral force on the Euclidean plane to constant curvature spaces. It is known that the angular momentum is not conserved in a noncentral setting; we take angular momentum and integral torque to be two independent coordinates and study two different reductions of noncentral forces using these two new variables. These lead to the curvature-dependent generalized Emden–Fowler and generalized Lane–Emden equations. These reduce to the standard form of the Emden–Fowler and Lane–Emden equations when the curvature vanishes. We compute all the generalized Emden–Fowler equations based on polynomial noncentral forces. Finally, we also give a brief outline of our construction for nonpolynomial forces.

The Calogero-Leyvraz Lagrangian framework, associated with the dynamics of a charged particle moving in a plane under the combined influence of a magnetic field as well as a frictional force, proposed by Calogero and Leyvraz, has some special features. It is endowed with a Shannon "entropic" type kinetic energy term. In this paper, we carry out the constructions of the 2D Lotka-Volterra replicator equations and the N=2 Relativistic Toda lattice systems using this class of Lagrangians. We take advantage of the special structure of the kinetic term and deform the kinetic energy term of the Calogero-Leyvraz Lagrangians using the κ-deformed logarithm as proposed by Kaniadakis and Tsallis. This method yields the new construction of the κ-deformed Lotka-Volterra replicator and relativistic Toda lattice equations.

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.

The position-dependent non-conservative forces are called curl forces introduced recently by Berry and Shukla (J Phys A 45:305201, 2012). The aim of this paper is to study mainly the curl force dynamics of non-conservative central force $$\ddot{x} = -xg(x,y)$$ and $$\ddot{y} = -yg(x,y)$$ connected to higher-order saddle potentials. In particular, we study the dynamics of the type $$\ddot{x}_i = -x_ig \big (\frac{1}{2}(x_{1}^{2} - x_{2}^{2}) \big )$$ , $$i=1,2$$ and its application towards the trapping of ions. We also study the higher-order saddle surfaces, using the pair of higher-order saddle surfaces and rotated saddle surfaces by constructing a generalized rotating shaft equation. The complex curl force can also be constructed by using this pair. By the direct computation, we show that all these motions of higher-order saddles are completely integrable due to the existence of two conserved quantities, viz. energy function and the Fradkin tensor. The Newtonian system $$\ddot{x} = {{\mathcal {X}}}(x,y)$$ , $$\ddot{y} = {{\mathcal {Y}}}(x,y)$$ has also been studied with an energy like first integral $$I(\mathbf{x}, \dot{\mathbf{x}}) = \frac{1}{2}\dot{\mathbf{x}}^TM(\mathbf{x})\dot{\mathbf{x}} + U(\mathbf{x})$$ , where $$M(\mathbf{x})$$ is a $$(2 \times 2)$$ matrix of which the components are polynomials of degree less than or equal to two and the condition on $${{\mathcal {X}}}$$ and $${{\mathcal {Y}}}$$ for which the curl is non-vanishing is also obtained.

In this paper, we generalize the application of the Levi-Civita (L-C) and Kustaanheimo–Stiefel (K-S) regularization methods to quantum mechanical systems in two and three dimensions. Schrödinger equations in two and three dimensions, describing a particle moving under the combined influence of [Formula: see text] and [Formula: see text] potentials are mapped to that of a harmonic oscillator with inverted sextic potential, and interactions, in two and four dimensions, respectively. Using the perturbative solutions of the latter systems, we derive the eigen spectrum of the former systems. Using Bohlin–Sundmann transformation, a mapping between the Schrödinger equations describing shifted harmonic oscillator and H-atom is also derived. Exploiting this equivalence, the solution to the former is obtained from the solution of the latter.

Noncentral force is an exemplary example of curl force and in general this is not an integrable system. The purpose of this note is twofold. First, we study a different reduction in the noncentral force compared to Berry and Shukla, and this leads to the generalized Emden–Fowler (GEF) equation, which in turn can be mapped to the Thomas–Fermi equation. Second, we compute the first integrals of the integrable standard Emden–Fowler (EF) and the generalized EF equations associated with the reduced noncentral dynamics using old results and new techniques. Finally, we compute the reduction in the nonpolynomial noncentral forces, which also leads to generalized EF equations.

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

In this present communication, the integrable modulation problem has been applied to study parametric extension of the Kapitza rotating shaft problem, which is a prototypical example of curl force as formulated by Berry and Shukla in (JPA 45:305201, 2012) associated with simple saddle potential. The integrable modulation problems yield parametric time-dependent integrable systems. The Hamiltonian and first integrals of the linear and nonlinear parametric Kapitza equation (PKE) associated with simple and monkey saddle potentials have been given. The construction has been illustrated by choosing $$ \omega (t)=a +b\cos t$$ and that maps to Mathieu-type equations, which yield Mathieu extension of PKE. We study the dynamics of these equations. The most interesting finding is the mixed mode of particle trapping and escaping via the heteroclinic orbits depicted with the parametric Mathieu–Kapitza equation, which are absent in the case of nonparametric cases.

In this present communication the relativistic formulation of the curl forces with saddle potentials has been performed. In particular, we formulated the relativistic version of the Kapitza equation. The dynamics and trapping phenomena of this equation have been studied both theoretically and numerically. The numerical results show interesting characteristics of the charged particles associated with the particle trapping and escaping in the relativistic domain. In addition, the relativistic generalization of the Kapitza equation associated with the monkey saddle has also been discussed.

Regularization of damped motion under central forces in two and three-dimensions are investigated and equivalent, undamped systems are obtained. The dynamics of a particle moving in $\frac{1}{r}$ potential and subjected to a damping force is shown to be regularized a la Levi-Civita. We then generalize this regularization mapping to the case of damped motion in the potential $r^{-\frac{2N}{N+1}}$. Further equation of motion of a damped Kepler motion in 3-dimensions is mapped to an oscillator with inverted sextic potential and couplings, in 4-dimensions using Kustaanheimo-Stiefel regularization method. It is shown that the strength of the sextic potential is given by the damping co-efficient of the Kepler motion. Using homogeneous Hamiltonian formalism, we establish the mapping between the Hamiltonian of these two models. Both in 2 and 3-dimensions, we show that the regularized equation is non-linear, in contrast to undamped cases. Mapping of a particle moving in a harmonic potential subjected to damping to an undamped system with shifted frequency is then derived using Bohlin-Sudman transformation., Comment: 19 pages, Introduction and Section 2 have overlap with our pre-print, arXiv:2102.06441

The position dependent nonconservative forces are called curl forces introduced recently by M.V. Berry, P. Shukla, J. Phys. A 45, 305201 (2012). In this article, we elucidate the role of curl forces in classical and quantum optics. At first we review the work of A. Ashkin and demonstrate the existence of curl force. We then discuss the role of curl force in ion trapping theory and map it to rotating saddle potential. Finally we show the connection between the two level atom problem, curl forces and their connection to Pais-Uhlenbeck oscillator.

This paper explores the class of equations of the Non-linear Schrodinger (NLS) type by employing both geometrical and spectral analysis methods. The work is developed in three stages. First, the geometrical method (AKS theorem) is used to derive different equations of the Non-linear Schrodinger (NLS) and Derivative Non-linear Schrodinger (DNLS) families. Second, the spectral technique (Tu method) is applied to obtain the hierarchies of equations belonging to these types. Third, the trace identity along with other techniques is used to obtain the corresponding Hamiltonian structures. It is found that the spectral method provides a simple algorithmic procedure to obtain the hierarchy as well as the Hamiltonian structure. Finally, the connection between the two formalisms is discussed and it is pointed out how application of these two techniques in unison can facilitate the understanding of integrable systems. In concurrence with Tu's method, Gesztesy and Holden also formulated a method of derivation of the trace formulas for integrable nonlinear evolution equations, this method is based on a contour-integration technique.

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.

Tulczyjew's triples are constructed for the Schmidt-Legendre transformations of both second and third-order Lagrangians. Symplectic diffeomorphisms relating the Ostrogradsky-Legendre and the Schmidt-Legendre transformations are derived. Several examples are presented.

In this paper a new approach to study an equation of the Lienard type with a strong quadratic damping is proposed based on Jacobi's last multiplier and Chiellini's integrability condition. We obtain a closed form solution of the transcedental characteristic equation of the Lienard type equation using the Lambert W-function.

For the first time, both non-holonomic and quasi-integrable deformations are obtained for the AB system of coupled equations. The AB system models geophysical and atmospheric fluid motion along with ultra-short pulse propagation in nonlinear optics and serves as a generalization of the well-known sine-Gordon equation. The non-holonomic deformation retains integrability subjected to higher-order differential constraints whereas the quasi-AB system, which is partially deviated from integrability, is characterized by an infinite subset of quantities (charges) that are conserved only asymptotically given the solution possesses definite space-time parity properties. Particular localized solutions to both these deformations of the AB system are obtained, some of which are qualitatively unique to the corresponding deformation, displaying similarities with physically observed excitations., Comment: 32 pages, 5 figures, affiliation updated, analysis extended, references added and funding added

This paper is concerned with the monotonicity of the period function for closed orbits of systems of the Lienard II type equation given by $${\ddot{x}} + f(x){\dot{x}}^{2} + g(x) = 0$$ . We generalize Chicone’s result regarding the monotonicity of the period function to planar Hamiltonian vector fields in the presence of a position dependent mass. Sufficient conditions are also given for the isochronicity of the potential in case of such a system.

Using a novel transformation involving the Jacobi Last Multiplier (JLM) we derive an old integrability criterion due to Chiellini for the Lienard equation. By combining the Chiellini condition for integrability and Jacobi's Last Multiplier the Lagrangian and Hamiltonian of the Lienard equation is derived. We also show that the Kukles equation is the only equation in the Lienard family which satisfies both the Chiellini integrability and the Sabatini criterion for isochronicity conditions. In addition we examine this result by mapping the Lienard equation to a harmonic oscillator equation using tacitly Chiellini's condition. Finally we provide a metriplectic and complex Hamiltonian formulation of the Lienard equation through the use of Chiellini condition for integrability.

The first integrals of the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh-Rose model, and the Oregonator model are derived using the method of Darboux polynomials. It is shown that, the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh-Rose model can be written in a bi-Hamiltonian/Nambu metriplectic form.

We introduce the Schlesinger transformations for the Gambier, linearisable, equation and by combining the former construct the contiguity relations of the solutions of the latter. We extend the approach to the discrete domain obtaining thus the Schlesinger transformations and the contiguity relations of the solutions of the Gambier mapping. In all cases the resulting contiguity relation is a linearisable equation, involving free functions, and which can be related to the generic Gambier mapping.

This article proposes a new two-parameter generalized entropy, which can be reduced to the Tsallis and Shannon entropies for specific values of its parameters. We develop a number of information-theoretic properties of this generalized entropy and divergence, for instance, the sub-additive property, strong sub-additive property, joint convexity, and information monotonicity. This article presents an exposit investigation on the information-theoretic and information-geometric characteristics of the new generalized entropy and compare them with the properties of the Tsallis and Shannon entropies.

We study scalar field theory as a generalization of point particle mechanics using the Polyakov action, and demonstrate how to extend Lorentzian and Riemannian Eisenhart lifts to the theory in a similar manner. Then we explore extension of the Randers-Finsler formulation and its principles to the Nambu-Goto action, and describe a Jacobi Lagrangian for it., Comment: 9 pages, Comments are most welcome

We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Li\'enard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov-Lewis invariant for a time dependent oscillator to (coupled) Li\'enard and Li\'enard type equations. We also study the linearization problem for the coupled Li\'enard equation using nonlocal transformations and derive coupled dissipative Ermakov-Milne-Pinney equation. As an offshoot of this nonlocal transformation method when the standard Li\'enard equation, x + f(x)x_ + g(x) = 0, is mapped to that of the linear harmonic oscillator equation we obtain a relation between the functions f(x) and g(x) which is exactly similar to the condition derived in the context of isochronicity of the Li\'enard equation., Comment: To appear in International Journal of Geometric Methods in Modern Physics

We show the connection between the theory of Hamiltonian curl forces introduced by Berry and Shukla and the cofactor systems introduced by the Linkoping school. The linear curl forces studied by Berry and Shukla are the dynamics related to saddle potentials. We also discuss the rotating saddle potential and its connection to the Bateman-like Lagrangian for a pair of damped and anti-damped oscillators. Here the pair of frictional terms can be mapped to the Coriolis-like force caused by the rotation of the potential as shown by Kirillov and Levi. We study the geometrical structure of the linear curl force. A hidden holomorphic structure is uncovered in this paper.

The method of Lie symmetries and the Jacobi Last Multiplier is used to study certain aspects of nonautonomous ordinary differential equations. Specifically we derive Lagrangians for a number of cases such as the Langmuir–Blodgett equation, the Langmuir–Bogulavski equation, the Lane–Emden–Fowler equation and the Thomas–Fermi equation by using the Jacobi Last Multiplier. By combining a knowledge of the last multiplier together with the Lie symmetries of the corresponding equations we explicitly construct first integrals for the Langmuir–Bogulavski equation q ¨ + 5 3 t q - t - 5 / 3 q - 1 / 2 = 0 and the Lane–Emden–Fowler equation. These first integrals together with their corresponding Hamiltonains are then used to study time-dependent integrable systems. The use of the Poincare–Cartan form allows us to find the conjugate Noetherian invariants associated with the invariant manifold.

Using Okamoto’s folding transformation we investigate the mapping of the Gambier equation and its higher-order analogs to the generalized Abel chain of equations. In particular we show how the Darboux polynomials and first integral of the Abel equation can be mapped to an equation of the Gambier type using folding transformations.

In this pedagogic article we study the geometrical structure of nonholonomic system and elucidate the relationship between Jacobi’s last multiplier (JLM) and nonholonomic systems endowed with the almost symplectic structure. In particular, we present an algorithmic way to describe how the two form and almost Poisson structure associated to nonholonomic system, studied by L. Bates and his coworkers (Rep Math Phys 42(1–2):231–247, 1998; Rep Math Phys 49(2–3):143–149, 2002; What is a completely integrable nonholonomic dynamical system, in Proceedings of the XXX symposium on mathematical physics, Torun, 1998; Rep Math Phys 32:99–115, 1993), can be mapped to symplectic form and canonical Poisson structure using JLM. We demonstrate how JLM can be used to map an integrable nonholonomic system to a Liouville integrable system. We map the toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system.

Through the Hasimoto map, various dynamical systems can be mapped to different integrodifferential generalizations of Nonlinear Schrodinger (NLS) family of equations some of which are known to be integrable. Two such continuum limits, corresponding to the inhomogeneous XXX Heisenberg spin chain [Balakrishnan, J. Phys. C 15, L1305 (1982)] and that of a thin vortex filament moving in a superfluid with drag [Shivamoggi, Eur. Phys. J. B 86, 275 (2013) 86; Van Gorder, Phys. Rev. E 91, 053201 (2015)], are shown to be particular non-holonomic deformations (NHDs) of the standard NLS system involving generalized parameterizations. Crucially, such NHDs of the NLS system are restricted to specific spectral orders that exactly complements NHDs of the original physical systems. The specific non-holonomic constraints associated with these integrodifferential generalizations additionally posses distinct semi-classical signature., Comment: 15 pages, 1 figure; to appear in EPJB

We demonstrate a generalization of quantum discord using a generalized definition of von-Neumann entropy, which is Sharma-Mittal entropy; and the new definition of discord is called Sharma-Mittal quantum discord. Its analytic expressions are worked out for two qubit quantum states as well as Werner, isotropic, and pointer states as special cases. The R{\'e}nyi, Tsallis, and von-Neumann entropy based quantum discords can be expressed as limiting cases for of Sharma-Mittal quantum discord. We also numerically compare all these discords and entanglement negativity., Comment: This article is similar to the one published in Quantum Information Processing (2019) 18 (6), 169

In this paper we employ three recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a family of so-called short-pulse equations (SPE). A recent, novel application of phase-plane analysis is first employed to show the existence of breaking kink wave solutions in certain parameter regimes. Secondly, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic (heteroclinic) orbits of the traveling-wave equations for the SPE equation, as well as for its generalized version with arbitrary coefficients. These correspond to pulse (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the SPE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized SPE equations considered here, the results obtained are both new and timely., Comment: accepted for publication in Communications in Nonlinear Science and Numerical Simulation

We study Hamiltonian analysis of three-dimensional advection flow $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ of incompressible nature $\nabla \cdot {\bf v} ={\bf 0}$ assuming that dynamics is generated by the curl of a vector potential $\mathbf{v} = \nabla \times \mathbf{A}$. More concretely, we elaborate Nambu-Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity $\mathbf{A} \cdot \nabla \times \mathbf{A}$. We present an example (satisfying $\mathbf{A} \cdot \nabla \times \mathbf{A} \neq 0$) which can be written as in the form of Nambu-Hamiltonian and bi-Hamiltonian formulations. We present another example (satisfying $\mathbf{A} \cdot \nabla \times \mathbf{A} = 0$) which we cannot able to write it in the form of a Nambu-Hamiltonian or bi-Hamiltonian system. On the hand, this second example can be manifested in terms of Hamiltonian one-form and yields generalized or vector Hamiltonian equations $\dot{x}_i = - \epsilon_{ijk}{\partial \eta_j}/{\partial x_k}$.

We obtain the first integrals of various extensions of the Mathieu equation by exploiting the integrable time-dependent classical dynamics introduced by Bartuccelli and Gentile (2003) [6]. We also compute the Lagrangian of the Van der Pol-Mathieu equation using Jacobi's last multiplier and consider certain coupled versions of time-dependent equations of the oscillator type.

We study second-order, second-degree systems related to the Painleve equations which possess one and two parameters. In every case we show that by introducing a quantity related to the canonical Hamiltonian variables it is possible to derive such a second-degree equation. We investigate also the contiguity relations of the solutions of these higher-degree equations. In most cases these relations have the form of correspondences, which would make them non-integrable in general. However, as we show, in our case these contiguity relations are indeed integrable mappings, with a single ambiguity in their evolution (due to the sign of a square root).

We study the construction of singular Lagrangians using Jacobi’s last multiplier (JLM). We also demonstrate the significance of the last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonian of the Host-Parasite model and a Lotka-Volterra mutualistic system, both of which are well known first-order systems of differential equations arising in biology.

The relativistic Lagrangian in presence of potentials was formulated directly from the metric, with the classical Lagrangian shown embedded within it. Using it we formulated covariant equations of motion, a deformed Euler-Lagrange equation, and relativistic Hamiltonian mechanics. We also formulate a modified local Lorentz transformation, such that the metric at a point is invariant only under the transformation defined at that point, and derive the formulae for time-dilation, length contraction, and gravitational redshift. Then we compare our formulation under non-relativistic approximations to the conventional ad-hoc formulation, and we briefly analyze the relativistic Lienard oscillator and the spacetime it implies., Comment: 24 pages

We present a construction of the Jacobi-Maupertuis (JM) principle for an equation of the Lienard type, $$ \ddot{x} + f(x) \dot{x}^2 + g(x) = 0, $$ using Jacobi's last multiplier. The JM metric allows us to reformulate the Newtonian equation of motion for a variable mass as a geodesic equation for a Riemannian metric. We illustrate the procedure with examples of Painleve-Gambier XXI, the Jacobi equation and the Henon-Heiles system.

Here we discuss first integrals of a particular representation associated with second-order ordinary differential equations. The linearization problem is a particular case of the equivalence problem together with a number of related problems such as defining a class of transformations, finding invariants of these transformations, obtaining the equivalence criteria, and constructing the transformation. The relationship between the integral form, the associated equations, equivalence transformations, and some examples are considered as part of the discussion illustrating some important aspects and properties.

We introduce a new transformation (nonlocal) to find the general solutions of some equations belonging to the third and fourth-order time dependent Riccati class of equations. These are in turn related to the Chazy polynomial class and the time dependent F-XVI Bureau symbol PI equations respectively.

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels [M.E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348, 1996, 5007–5029], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi–Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painleve test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.

We derive the Lagrangians of the higher-order Painleve equations using Jacobi’s last multiplier technique. Some of these higher-order differential equations display certain remarkable properties like passing the Painleve test and satisfy the conditions stated by Juras [M. Juras, The inverse problem of the calculus of variations for sixth- and eighth-order scalar ordinary differential equations, Acta Appl. Math. 66 (1) (2001) 25–39], thus allowing for a Lagrangian description.

The influence of barium oxide, heat treatment time and temperature on the crystallization, microstructure and mechanical behavior of the system Bax·K1−2x·Mg3·Al·Si3O10·F2 (where x = 0.0, 0.3 and 0.5) was investigated in order to develop novel, high strength and machinable glass–ceramics. Three glasses were prepared and characterized by differential thermal analysis (DTA), X-ray diffraction (XRD), scanning electron microscope (SEM) techniques and some mechanical testing methods. The crystallization kinetics of glass–ceramics was also studied. Activation energy and Avrami exponent calculated for the crystallization peak temperature (Tp) of three different glass batches. The Vickers hardness decreased slightly on formation of the potassium fluorophlogopite and barium fluorophlogopite phases, but decreased significantly on formation of an interconnected ‘house of cards’ microstructure.