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The purpose of this paper is to illustrate the theory and methods of analytical mechanics that can be effectively applied to the research of some nonlinear nonconservative systems through the case study of two-dimensionally coupled Mathews-Lakshmanan oscillator (abbreviated as M-L oscillator). (1) According to the inverse problem method of Lagrangian mechanics, the Lagrangian and Hamiltonian function in the form of rectangular coordinates of the two-dimensional M-L oscillator is directly constructed from an integral of the two-dimensional M-L oscillators. (2) The Lagrange and Hamiltonian function in the form of polar coordinate was rewritten by using coordinate transformation. (3) By introducing the vector form variables, the two-dimensional M-L oscillator motion differential equation, the first integral, and the Lagrange function are written. Therefore, the two-dimensional M-L oscillator is directly extended to the three-dimensional case, and it is proved that the three-dimensional M-L oscillator can be reduced to the two-dimensional case. (4) The two direct integration methods were provided to solve the two-dimensional M-L oscillator by using polar coordinate Lagrangian and pointed out that the one-dimensional M-L oscillator is a special case of the two-dimensional M-L oscillator.

Nonlinear systems are not only the traditional research fields of mechanical physics and engineering science but also the research fields of other natural science and many social science [

Some theoretical studies of nonlinear dynamic systems require the derivation of Lagrange functions and Hamiltonian functions corresponding to these equations and even require accurate analytical solutions. For example, in 1974, Mathews and Lakshmanan obtained a M-L oscillator subsystem by deformation of the linear conservative oscillator subsystem, which is a nonlinear nonconservative oscillator subsystem, but has a strictly controlled periodic solution [

This article will systematically use analytical mechanics theory and methods to study two-dimensional M-L oscillators. First, after obtaining two integrals with clear physical meanings of the oscillator equation, a two-dimensional M-L oscillator Lagrangian function and Hamiltonian function are constructed directly from an integral according to the theory and method of Lagrangian mechanics inverse problems. And the Lagrange function and Hamiltonian function in the form of plane polar coordinates are derived by using variable transformation [

In 1974, Mathews and Lakshmanan introduced a one-dimensional M-L vibrator:

The one-dimensional M-L oscillator (

Equation (

Integral

According to the Lagrange mechanics, the differential equation of the system can be derived if the Lagrange equation of the system is known. The inverse problem of the Lagrange mechanics is the differential equation of the system that is known, and it is used to test whether this equation can be written into the form of the Lagrange equation and how to write it in such way [

If we set the differential equation of the system motion as

The undetermined factors

The Lagrange function of the two-dimensional M-L oscillator (

Select the energy integral

Substituting this into Equation (

When the plane polar coordinates of

In the form of plane polar coordinates, kinematic integrals are expressed as

In Equations (

According to the Lagrange 3quation (

The inverse solution are

The Hamiltonian equation of the two-dimensional M-L oscillator is obtained by means of Legendre transformation:

Similarly, the generalized momentum and Hamilton equation corresponding to plane polar coordinates can be derived as

The vector of position is introduced from the rectangular coordinates of

The position vector is hence introduced as

The differential equation (

Correspondingly, the first integrals Equations (

In the form of vector variable, the Lagrange equation (

If the M-L oscillator is extended to the three-dimensional case, the two-dimensional plane potential vector of Equation (

Therefore, Equations (

Correspondingly, there are three component conservation formulas for the angular momentum integral (

It should be noted that the extension of the three-dimensional case of the M-L oscillator has no special significance. According to the previous discussion, the expression (

In such a coordinate system, Equation (

After Obtaining the Lagrange function in polar form, the two-dimensional M-L oscillator can be solved directly by two methods:

The Lagrangian function of the radial motion of the oscillator can be derived by using Equations (

Now the motion of the two-dimensional M-L oscillator has been reduced to the problem of one-dimensional radial motion, which can be directly integrated.

As can be seen from Equation (

An orbital differential equation of a two-dimensional M-L oscillator can be derived by substituting the energy-type integral (

By integrating this equation, the orbit equation of the oscillator (

In the study of modern physics, for example, it may involve the quantization of some nonconservative nonlinear systems when dealing with some mesoscopic physical systems. At this time, it is often necessary to derive the Hamiltonian function of the system. The Hamiltonian function of the two-dimensional M-L oscillator derived in this paper can be directly solved by Hamiltonian mechanics methods, such as Hamilton-Jacobi theory; on the other hand, it also lays the foundation for further discussion of the quantization of this system.

There are many ways to study nonlinear and nonconservative systems, but analytical mechanics theories and methods have an important value in such research. In some studies, e.g., discussing the quantization problem, it is necessary to first mechanize the system analysis; that is to say, the Lagrange and Hamiltonian functions of the system need to be derived first. In this paper, according to the Lagrange mechanics inverse problem theory and method, the Lagrange function and Hamiltonian function of the oscillator can be constructed directly from the energy form integral of the two-dimensional M-L oscillator, which can be used to realize the analytical mechanics of the nonlinear and nonconservative system. The two-dimensional M-L vibrator is solved by using traditional analytical mechanics methods such as coordinate transformation and motion integration reduction order separation variables. At the same time, the relationship between the two-dimensional M-L oscillator and the one-dimensional and three-dimensional M-L oscillator is discussed. By studying the analytical mechanics of the two-dimensional M-L oscillator, it is shown that the methodology of analytical mechanics is of vital value in the study of some nonlinear nonconservative systems. The key lies in the analysis and mechanization of nonlinear differential equations of motion, meaning that the Lagrange function and Hamilton function of nonlinear equations can be derived by using analytical mechanics inverse problem methodology.

No data were used to support this study of “The Lagrangian and Hamiltonian for the Two-dimensional Mathews-Lakshmanan Oscillator.”

The authors declare no conflict of interest.

The authors gratefully acknowledge the financial support by Natural Science Research Project of Anhui Education Department (PX-271191861) and School Enterprise Cooperation Practical Education Base of Anhui Education Department (2017sjjd050).