Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Song, Chuan-Jing; Cheng, Yao

doi:https://doi.org/10.1155/2021/1959643

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Applications Results and Discussion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 1959643 | https://doi.org/10.1155/2021/1959643

Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Chuan-Jing Song¹and Yao Cheng¹

Academic Editor: Ivan Giorgio

Received22 Aug 2021

Accepted13 Sept 2021

Published29 Sept 2021

Abstract

Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.

1. Introduction

With the development of discipline and the progress of technology, the dynamics of the constrained mechanical systems was put forward, so Analytical Mechanics appears. Lagrange is the founder of Analytical Mechanics. Lagrange further studied the motion of the constrained particles after d’Alembert. The Lagrange equation and the d’Alembert-Lagrange principle are the core of Lagrangian mechanics.

Hamilton developed Analytical Mechanics. In his two long papers, “On a General Method in Dynamics” (1834) and “Second Essay on a General Method in Dynamics” (1835), he proposed an integral variational principle and a dynamic equation with generalized coordinate and generalized momentum as independent variables. This principle is called the Hamilton principle. The dynamic equation given by Hamilton is called the canonical equation. The Hamilton principle and the Hamilton canonical equation are the core of Hamiltonian mechanics.

Hamilton’s principle is highly general, which can represent the motion law of a holonomic and conservative system by only one functional extreme value. The principle is not only simple and beautiful in form but also rich and profound in connotation. It can be applied to mechanics, optics, electromagnetism, and other fields and can also be applied to approximate calculation [1, 2]. In Ref. [3], the Hamilton principle is applied to the dynamics of the flexible multibody and rigid flexible coupling systems. Hamilton’s principle is extended to the holonomic nonconservative system [4], the high-order system [5], and the nonholonomic system [4]. In addition, the Pfaff-Birkhoff principle [6, 7], generalized Pfaff-Birkhoff principle [8], and Vujanović’s variational principle of nonconservative system [9] are also the generalization of Hamilton’s principle.

The Hamilton equation is also of great significance. Firstly, the canonical equation is simpler in form and more symmetrical in structure than the Lagrange equation. It is more convenient for general discussion when solving many complex mechanical problems, such as celestial mechanics and vibration theory. Secondly, the new concepts related to the canonical equations, such as the canonical variables, have many applications in mechanics and physics, such as statistical physics and quantum mechanics. Thirdly, a complete set of integration methods are established for the canonical equations, such as the Poisson theorem, Jacobi method, canonical transformation, and integral invariants [4]. Fourthly, geometric mechanics has been developed due to the symplectic structure of the canonical equations. Besides, Hamiltonian mechanics has also contributed to the formation and the development of the generalized Hamiltonian mechanics [10] and the Birkhoffian mechanics [11]. Finally, considering the perturbation of the Hamilton equation, the KAM theorem appears. The KAM theorem became the beginning of the chaos theory [4].

In 1918, the famous paper “Invariante Variations Probleme” by German mathematician Noether revealed the relationship between the conserved quantity of the mechanics system and its internal dynamical symmetry [12]. Reference [13] points out that the application of the mechanics variational principle and its physical significance are based on two theorems: the Hilbert independence theorem and the Noether theorem. The first theorem gives the mathematical argument of the variational principle, and the second theorem reveals its physical significance.

The Noether symmetry method is one of the modern integration methods of the Hamiltonian mechanics. For the Hamiltonian system, Noether symmetry is the invariance of the Hamilton action under the infinitesimal transformations. The Noether symmetry method points out that if the infinitesimal generators and the gauge function satisfy the Noether identity, then the conserved quantity of the system can be found [14–20]. The advantage of the Noether theory is that if there is a Noether symmetry, a corresponding conserved quantity can be found and vice versa. The Noether theory can also be used to solve general ordinary differential equations as long as they are expressed as equations of mechanics systems [21].

Recent developments in the fields of science, engineering, economics, bioengineering, and applied mathematics have demonstrated that many phenomena in nature are modelled more accurately using fractional derivatives [22, 23]. Fractional Hamiltonian mechanics [24–32] and Noether theorems for them have also been established and investigated [33–36]. The fractional operators mainly referred to the left and right fractional Riemann-Liouville integrals, the left and right fractional Riemann-Liouville differential operators, the left and right fractional Caputo differential operators, the Riesz-Riemann-Liouville differential operator, and the Riesz-Caputo differential operator. However, Agrawal [37] pointed out that the fractional power kernel need not be the only kernel to describe the phenomena of the nature. He defined three new operators which in special cases reduce to the fractional operators listed above. Then, the entire theories of the Hamiltonian mechanics and the corresponding Noether theorem can be redeveloped. And in such a case, the theories of the integer order and the fractional order Hamiltonian mechanics and other results resulted from them would be special cases of the more general new operators.

The structure of this paper is organized as follows. Section 2 lists the definitions and properties of the generalized operators briefly. Fractional calculus of variations for the Hamiltonian system within generalized operators is studied in Section 3. Based on the generalized operators, Noether symmetry, and conserved quantities, perturbation to Noether symmetry and adiabatic invariants are investigated in Section 4 and Section 5, respectively. Section 6 presents two applications to illustrate the methods and results obtained in this paper. A conclusion is given in Section 7.

2. Preliminaries

The definitions and properties of the generalized operators , , and were studied in detail by Agrawal [37]. Here, we only list their definitions and integration by part formulae.

The operators , , and have the forms where is continuous and integrable, , is a parameter set, and are two real numbers, is a kernel which may depend on a parameter , and is an integer.

Remark 1. Let , then different results will be obtained under different conditions. For example, when , we have i.e., the operator and the operator reduce to the left Riemann-Liouville fractional operator and the left Caputo fractional operator, respectively. When , we have i.e., the operator and the operator reduce to the right Riemann-Liouville fractional operator and the right Caputo fractional operator, respectively. When , we have i.e., the operator and the operator reduce to the Riesz-Riemann-Liouville fractional operator and the Riesz-Caputo fractional operator, respectively.

The generalized operators , , and satisfy the following integration by part formulae: where , , and is an integer.

In this text, we set , so . The variational problems within generalized operators and are to be studied first.

3. Hamilton Equations within Generalized Operators

3.1. Hamilton Equation within Generalized Operator

Let be the Lagrangian within generalized operator , , and , then the generalized momentum is defined as , , and the Hamiltonian can be expressed as . Therefore, the Hamilton action within generalized operator is

Then, with the commutative conditions [38]. and the boundary conditions is called Hamilton principle within generalized operator , where means the isochronous variation.

Using Equations (1), (8), and (12), we have

Then, from the Hamiltonian , the independence of , and the arbitrariness of the interval , we get

Equation (15) is called the Hamilton equation within generalized operator .

Remark 2. Let ; from Equation (15), we can get the Hamilton equations in terms of the left Riemann-Liouville fractional operator, the right Riemann-Liouville fractional operator, and the Riesz-Riemann-Liouville fractional operator by letting , , and , respectively.

3.2. Hamilton Equation within Generalized Operator

Let be the Lagrangian within the generalized operator , , and , then the generalized momentum is defined as , , and the Hamiltonian can be expressed as . Therefore, the Hamilton action within the generalized operator is

Then, with the commutative conditions [38] and the boundary conditions are called the Hamilton principle within generalized operator , where means the isochronous variation.

Using Equations (9), (17), and (19), we have

Then, from the Hamiltonian , the independence of , and the arbitrariness of the interval , we get

Equation (21) is called the Hamilton equation within generalized operator .

Remark 3. Equation (21) is consistent with the result obtained in Ref. [39]. However, the method in Ref. [39] is different.

Remark 4. Let ; from Equation (21), we can get the Hamilton equations in terms of the left Caputo fractional operator, the right Caputo fractional operator, and the Riesz-Caputo fractional operator by letting , , and , respectively.

Remark 5. In Remark 2 and Remark 4, there are six cases in total. If we let , then all of them reduce to the classical Hamilton equation, which can be found in Ref. [14].

4. Noether Theorems within Generalized Operators

Noether symmetry means the invariance of the Hamilton action. Noether symmetry leads to conserved quantity. Therefore, the change of the Hamilton action under the infinitesimal transformations will be studied. There are also two parts: one is the Noether symmetry and conserved quantity in terms of operator , and the other is in terms of operator . We begin with the definition of conserved quantity.

Definition 6. A quantity is called a conserved quantity if and only if the condition holds.

4.1. Noether Theorem within Generalized Operator

First, we give the infinitesimal transformations within generalized operator as

Expanding Equation (22), we have where is an infinitesimal parameter and , , and are called infinitesimal generators within the generalized operator .

Then, let be the linear part of and neglecting the higher order of , we get where

It follows from Noether symmetry () that

Equation (26) is called the Noether identity within the generalized operator .

Finally, the conserved quantities within generalized operators deduced by the Noether symmetry are presented.

Theorem 7. For the Hamiltonian system within generalized operator (Equation (15)), if the infinitesimal generators , , and satisfy the Noether identity (Equation (26)), then there exists a conserved quantity

Proof. From Equations (15) and (26), we have .

If we let , , and be called the gauge function, then from Equation (24), we have

Equation (28) is called the Noether quasi-identity within the generalized operator .

In this case, we have the following theorem.

Theorem 8. For the Hamiltonian system within the generalized operator (Equation (15)), if there exists a gauge function such that the infinitesimal generators , , and satisfy the Noether quasi-identity (Equation (28)), then there exists a conserved quantity

Proof. From Equations (15) and (28), we have .

Remark 9. Let ; from Equations (26) and (28), Theorem 7, and Theorem 8, we can get Noether identities, Noether quasi-identities, and Noether theorems in terms of the left Riemann-Liouville fractional operator, the right Riemann-Liouville fractional operator, and the Riesz-Riemann-Liouville fractional operator by letting , , and , respectively.

4.2. Noether Theorem within Generalized Operator

The infinitesimal transformations are and the expanded forms of the infinitesimal transformations are where is an infinitesimal parameter and , , and are called infinitesimal generators within the generalized operator .

Similarly, let be the linear part of and neglecting the higher order of , we get where

Letting , we have

Equation (34) is called the Noether identity within the generalized operator .

Let , , and be called the gauge function, then from Equation (32), we have

Equation (35) is called the Noether quasi-identity within the generalized operator .

Therefore, we have the following theorem.

Theorem 10. For the Hamiltonian system within the generalized operator (Equation (21)), if the infinitesimal generators , , and satisfy the Noether identity (Equation (34)), then there exists a conserved quantity

Proof. From Equations (21) and (34), we have .

Theorem 11. For the Hamiltonian system within the generalized operator (Equation (21)), if there exists a gauge function such that the infinitesimal generators , , and satisfy the Noether quasi-identity (Equation (35)), then there exists a conserved quantity

Proof. From Equations (21) and (35), we have .

Remark 12. Let ; from Equations (34) and (35), Theorem 10, and Theorem 11, we can get Noether identities, Noether quasi-identities, and Noether theorems in terms of the left Caputo fractional operator, the right Caputo fractional operator, and the Riesz-Caputo fractional operator by letting , , and , respectively. Particularly, the results obtained when are consistent with the results presented in Ref. [33].

Remark 13. In Remark 9 and Remark 12, there are six cases in total. If we let , then all of them reduce to the classical Noether identity, the classical Noether quasi-identity, and the classical Noether theorem, which can be found in Ref. [14].

5. Adiabatic Invariants within Generalized Operators

This section begins with the definition of the adiabatic invariant.

Definition 14. A quantity is called an adiabatic invariant if contains a parameter , whose highest power is , and also satisfies that is in proportion to .

When the systems (Equations (15) and (21)) are disturbed by small forces, the conserved quantities may also change.

Assuming that the Hamiltonian system within the generalized operator (Equation (15)), the gauge function and the infinitesimal generators , , and are disturbed as then we have the following theorem.

Theorem 15. For the disturbed Hamiltonian system within the generalized operator (Equation (38)), if there exists a gauge function such that the infinitesimal generators , , and satisfy where when , then there exists an adiabatic invariant

Proof. From Equations (38) and (40), we have .

If the Hamiltonian system within the generalized operator (Equation (21)), the gauge function and the infinitesimal generators , , and are disturbed as then we have the following theorem.

Theorem 16. For the disturbed Hamiltonian system within the generalized operator (Equation (42)), if there exists a gauge function such that the infinitesimal generators , , and satisfy where when , then there exists an adiabatic invariant

Proof. From Equations (42) and (46), we have .

Remark 17. Letting , we can get the adiabatic invariants in terms of the left Riemann-Liouville fractional operator, the right Riemann-Liouville fractional operator, and the Riesz-Riemann-Liouville fractional operator from Theorem 15 by letting , , and , respectively. Similarly, we can get the adiabatic invariants in terms of the left Caputo fractional operator, the right Caputo fractional operator, and the Riesz-Caputo fractional operator by letting , , and , respectively. Particularly, the adiabatic invariant within the Riesz-Caputo fractional operator is consistent with the result obtained in Ref. [33].

Remark 18. In Remark 17, there are six cases in total. If we let , then all of them reduce to the classical adiabatic invariant, which can be found in Ref. [33].

Remark 19. The adiabatic invariants of Theorem 15 and Theorem 16 reduce to conserved quantities of Theorem 8 and Theorem 11 when , respectively.

In the following text, we give two applications to illustrate the results and methods.

6. Applications

6.1. Application 1

We try to find the conserved quantity and adiabatic invariant for the two-dimensional isotropic harmonic oscillator within the generalized operator , whose Lagrangian has the form where and are constants.

From Equation (48), we have

Then, Equation (15) gives

And under the condition , we can verify that satisfy the Noether quasi-identity (Equation (28)). Therefore, from Theorem 8, we have

When the system is disturbed by and , then we can find that is a solution to Equation (40). Therefore, from Theorem 15, we obtain

Specially, let and (or or ); further, letting , we have

6.2. Application 2

The Lotka biochemical oscillator model’s Hamiltonian has the form

We try to find its conserved quantity and adiabatic invariant based on generalized operator .

Equation (21) gives the Hamilton equation within the generalized operator

Then, under the condition , it is easy to verify that is a solution to the Noether quasi-identity (Equation (35)). Therefore, from Theorem 11, we have

When the system is disturbed by and , then we can find that is a solution to Equation (46). Therefore, from Theorem 16, we have

Specially, let and (or or ); further, letting , we have

7. Results and Discussion

Hamilton equations, Noether theorems, and adiabatic invariants are obtained for the Hamiltonian systems on the basis of the generalized operators. The Hamilton equation in terms of the generalized operator (Equation (15)), the Hamilton equation in terms of the generalized operator (Equation (21)), the Noether symmetries and conserved quantities in terms of the generalized operator (Theorem 7 and Theorem 8), the Noether symmetries and conserved quantities in terms of the generalized operator (Theorem 10 and Theorem 11), the perturbation to Noether symmetries and adiabatic invariants in terms of the generalized fractional operators (Theorem 15 and Theorem 16) are all new work.

However, we still need to make efforts to finish the follow-up work successfully. For example, MATLAB is a useful tool to simulate the results of the applications to directly illustrate the achievements obtained in this paper. Therefore, the appliance of MATLAB is necessary. Besides, except for the Noether symmetry method, the Lie symmetry method, and the Mei symmetry method are also significant methods to be studied for the Hamiltonian systems within generalized operators. Time scale calculus is recently proposed with tremendous potential for applications. The Hamiltonian mechanics within mixed time scale and generalized operators is a new research direction, which also deserves to be done.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Chuan-Jing Song wrote the first draft of the manuscript. All authors reviewed and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant numbers 11802193, 11801396, and 11972241), the Natural Science Foundation of Jiangsu Province (grant numbers BK20191454 and BK20170374), and the Young Scientific and Technological Talents Promotion Project of Suzhou Association for Science and Technology.

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Copyright © 2021 Chuan-Jing Song and Yao Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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