LAGRANGIANS FOR DISSIPATIVE NONLINEAR OSCILLATORS: THE METHOD OF JACOBI LAST MULTIPLIER

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to several equations, including a class of equations recently studied by Musielak with his own method [Z. E. Musielak, Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients J. Phys. A: Math. Theor. 41 (2008) 055205], and in particular a Liènard type nonlinear oscillator and a second-order Riccati equation. Also, we derive more than one Lagrangian for each equation.     

LAGRANGIANS FOR BIOLOGICAL MODELS

We show that a method presented in [S. L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol. 48 (1974) 299–324] and later in [G. H. Paine, The development of Lagrangians for biological models, Bull. Math. Biol. 44 (1982) 749–760] for finding Lagrangians of classic models in biology, is actually based on finding the Jacobi Last Multiplier of such models. Using known properties of Jacobi Last Multiplier we show how to obtain linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation that can be derived from them, even in the case where those authors failed such as the host-parasite model. Also we show that the Lagrangians of certain second-order ordinary differential equations derived by Volterra in [V. Volterra, Calculus of variations and the logistic curve, Hum. Biol. 11 (1939) 173–178] are particular cases of the Lagrangians that can be obtained by means of the Jacobi Last Multiplier. Actually we provide more than one Lagrangian for those Volterra's equations.     

Hamilton–Jacobi Treatment of Constrained Systems with Second-Order Lagrangians

A new approach is examined in this paper for solving mechanical problems for both constrained and unconstrained systems with second-order Lagrangians, using the Hamilton–Jacobi formulation. The relevant Hamilton–Jacobi function is constructed first. This is then used to determine the solutions of the equations of motion for both systems.     

Singular Lagrangian, Hamiltonization and Jacobi last multiplier for certain biological systems

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.     

BI-HAMILTONIAN REPRESENTATION,SYMMETRIES AND INTEGRALS OF MIXED HEAVENLY AND HUSAIN SYSTEMS

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain equation, formulated in a two-component form, we present recursion operators, Lax pairs of Olver–Ibragimov–Shabat type and discover their Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all point and second-order symmetries, integrals and bi-Hamiltonian representations of these systems and their symmetry flows together with infinite hierarchies of nonlocal higher symmetries.     

Quantization of Higher-Order Constrained Lagrangian Systems Using the WKB Approximation

A general theory is given for solving the Hamilton–Jacobi partial differential equations (HJPDEs) for both constrained and unconstrained systems with arbitrarily higher-order Lagrangians. The Hamilton–Jacobi function is obtained for both types of systems by solving the appropriate set of HJPDEs. This is used to determine the solutions of the equations of motion. The quantization of both systems is then achieved using the WKB approximation. In constrained systems, the constraints become conditions on the wave function to be satisfied in the semiclassical limit.     

关于Birkhoff表示的Lagrange像的研究

研究运动微分方程Birkhoff表示的Lagrange像.得出二阶Lagrange函数应满足的条件,在此条件下广义Lagrange方程为二阶微分方程组;提出新的求解Lagrange力学逆问题路线;指出在此问题研究中曾发生过的失误.举例说明所得结果的应用.     

Exact solutions of differential equations with delay for dissipative systems

Exact solutions of the first order differential equation with delay are derived. The equation has been introduced as a model of traffic flow. The solution describes the traveling cluster of jam, which is characterized by Jacobi's elliptic function. The induced differential-difference equations are related to some soliton systems.     

LEPAGE EQUIVALENTS OF SECOND-ORDER EULER–LAGRANGE FORMS AND THE INVERSE PROBLEM OF THE CALCULUS OF VARIATIONS

In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler–Lagrange equations. They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton–Jacobi theory and theory of integration of variational equations. In this paper, Lepage equivalents of second-order Euler–Lagrange quasi-linear PDE's are characterised explicitly. A closed (n + 1)-form uniquely determined by the Euler–Lagrange form is constructed, and used to find a geometric solution of the inverse problem of the calculus of variations.     

经典力学中加速度相关的Lagrange函数

研究了加速度线性相关的Lagrange函数,在加速度项系数对称的条件下,Lagrange方程保持为二阶微分方程;给出了从运动方程构造加速度相关的Lagrange函数的方法;研究同一系统的加速度相关和加速度无关的Lagrange函数之间的关系.举例说明结果的应用. 关键词： Lagrange方程 加速度相关的Lagrange函数 广义力学 Lagrange函数的规范变换     

Connecting Jacobi elliptic functions with different modulus parameters

The simplest formulas connecting Jacobi elliptic functions with different modulus parameters were first obtained over two hundred years ago by John Landen. His approach was to change integration variables in elliptic integrals. We show that Landen’s formulas and their subsequent generalizations can also be obtained from a different approach, using which we also obtain several new Landen transformations. Our new method is based on recently obtained periodic solutions of physically interesting non-linear differential equations and remarkable new cyclic identities involving Jacobi elliptic functions.     

Symmetry of Lagrangians of a holonomic variable mass system

The symmetry of Lagrangians of a holonomic variable mass system is studied.Firstly,the differential equations of motion of the system are established.Secondly,the definition and the criterion of the symmetry of the system are presented.Thirdly,the conditions under which there exists a conserved quantity deduced by the symmetry are obtained.The form of the conserved quantity is the same as that of the constant mass Lagrange system.Finally,an example is shown to illustrate the application of the result.     

Gravitational Field as a Pressure Force from Logarithmic Lagrangians and Non-Standard Hamiltonians:The Case of Stellar Halo of Milky Way

Recently,the notion of non-standard Lagrangians was discussed widely in literature in an attempt to explore the inverse variational problem of nonlinear differential equations.Different forms of non-standard Lagrangians were introduced in literature and have revealed nice mathematical and physical properties.One interesting form related to the inverse variational problem is the logarithmic Lagrangian,which has a number of motivating features related to the Li′enard-type and Emden nonlinear differential equations.Such types of Lagrangians lead to nonlinear dynamics based on non-standard Hamiltonians.In this communication,we show that some new dynamical properties are obtained in stellar dynamics if standard Lagrangians are replaced by Logarithmic Lagrangians and their corresponding non-standard Hamiltonians.One interesting consequence concerns the emergence of an extra pressure term,which is related to the gravitational field suggesting that gravitation may act as a pressure in a strong gravitational field.The case of the stellar halo of the Milky Way is considered.     

Hamilton–Jacobi Quantization of Singular Lagrangians with Linear Velocities

In this paper, constrained Hamiltonian systems with linear velocities are investigated by using the Hamilton–Jacobi method. The integrablity conditions are considered on the equations of motion and the action function as well in order to obtain the path integral quantization of singular Lagrangians with linear velocities.     

Lam函数和非线性演化方程的扰动方法

利用小扰动方法对非线性演化方程作展开得到原始方程的各级近似方程.应用Jacobi椭圆函 数展开法求得了零级近似方程的准确解，并由此得到一级近似方程和二级近似方程分别满足 齐次Lam方程和非齐次Lam方程，应用Lam函数和Jacobi椭圆函数展开法可以分别求得一级近似方程和二级近似方程的准确解.这样，就求得了非线性演化方程的多级准确解. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

A Geometric Approach to Integrability of Abel Differential Equations

A geometric approach is used to study the Abel first-order differential equation of the first kind. The approach is based on the recently developed theory of quasi-Lie systems which allows us to characterise some particular examples of integrable Abel equations. Second order Abel equations will be discussed and the inverse problem of the Lagrangian dynamics is analysed: the existence of two alternative Lagrangian formulations is proved, both Lagrangians being of a non-natural class. The study is carried out by means of the Darboux polynomials and Jacobi multipliers.     

CONSERVATION LAWS FOR SELF-ADJOINT FIRST-ORDER EVOLUTION EQUATION

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.     

A Universal Solution

Abstract

The phenomenon of an implicit function which solves a large set of second order partial differential equations obtainable from a variational principle is explicated by the introduction of a class of universal solutions to the equations derivable from an arbitrary Lagrangian which is homogeneous of weight one in the field derivatives. This result is extended to many fields. The imposition of Lorentz invariance makes such Lagrangians unique, and equivalent to the Companion Lagrangians introduced in [1].     

Simple explicit formulas for Gaussian path integrals with time-dependent frequencies

Quadratic fluctuations require an evaluation of ratios of functional determinants of second-order differential operators. We relate these ratios to the Green functions of the operators for Dirichlet, periodic and antiperiodic boundary conditions on a line segment. This permits us to take advantage of Wronski's construction method for Green functions without knowledge of eigenvalues. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2 × 2 determinant of a constant matrix constructed from two linearly independent solutions of the homogeneous differential equations associated with the second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of this classical solution. In the presence of a zero mode, our method allows for a simple universal regularization of the functional determinants. For Dirichlet's boundary condition, our result is equivalent to Gelfand-Yaglom's. Explicit formulas are given for a harmonic oscillator with an arbitrary time-dependent frequency.     

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the （2＋1）-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.