The Herglotz variational principle offers an effective method for studying nonconservative system dynamics. The aim of this paper is to study the conservation laws of nonholonomic systems by using the Herglotz type generalized variational principle and establish Noether’s theorem and its inverse theorem for this system. In deriving the equations of motion, we use the Suslov definition of the reciprocity relation between differential and variational operations. First, the Herglotz type generalized variational principle is listed, and the Herglotz type Chaplygin equations for nonconservative nonholonomic systems are deduced. Second, Noether’s theorem and Noether’s inverse theorem are established, and the Herglotz type conservation laws are given. Finally, an example is provided to illustrate the practical implementation of the findings.
I. INTRODUCTION
Hertz1 divided constraints into holonomic and nonholonomic categories for the first time. Since then, nonholonomic mechanics have become an important branch of analytical mechanics. In recent years, there has been a notable expansion of nonholonomic mechanics into other fields such as geometric mechanics,2–4 robot dynamics,5–8 and motion planning and control.9–11 This has facilitated interdisciplinary research and application collaborations with other relevant subjects. The commutative property of differentiation operation d and variation operation δ is one of the fundamental problems in the study of nonholonomic systems.12 Historically, there have been two opposing viewpoints12,13 regarding the commutativity of d and δ. The first view is that regardless of whether the system is holonomic or nonholonomic, both are commutative, represented by Hölder and Voronetz. In contrast, the second view holds that d and δ are interchangeable only for holonomic systems, represented by Suslov and Levi-Civita.
The study of conservation laws in dynamical systems is extremely important both mathematically and physically. Conservation laws have multiple important roles, such as, first, when the differential equations of motion of a system are difficult to solve, the existence of conservation laws can at least partially understand the physical or dynamic behavior of the system. The conservation laws can be used to reduce the differential equation and simplify the solution of the problem greatly. Finally, the stability of the dynamical system can be analyzed by means of conservation laws.
Contemporary approaches of finding conservation laws usually involve studying the invariance of action or dynamical functions under infinitesimal transformations, which can also be explored through the differential variational principle. The classical variational principle has served as a fundamental cornerstone in the advancement of various disciplines, such as physics, mathematics, mechanics, and other fields. However, it no longer works in many important systems, such as nonconservative systems. Herglotz14 proposed a new generalized variational principle, which can be used to describe nonconservative systems including mechanical, thermodynamic, and quantum systems.15–19 The Herglotz variational principle has broader properties. It skillfully describes all the physical processes covered by Hamilton’s principle, while excelling at solving nonconservative challenges where Hamilton’s principles no longer apply. When the Lagrangian does not depend on the functional z, it degenerates to the classical integral variational principle. Georgieva first studied the Herglotz type Noether theorem with one independent variable20 and extended it to encompass several independent variables.21 Santos et al. studied the Herglotz type Noether theorems for the optimal control problem,22 the high-order variational problem,23,24 and with time-delays.25,26 So far, the Herglotz type Noether theorems for Hamilton systems, Birkhoff systems, and nonholonomic systems have been constructed in Refs. 27–32. In Ref. 30, nonholonomic systems are studied under the Hölder definition of commutative relations between differential and variational operations. In this paper, our focus will be on investigating the Herglotz type conservation laws of nonconservative nonholonomic systems by using the Suslov definition.
The structure of this article is outlined as follows: Sec. II is the theoretical part and consists of four subsections. In Subsection II A, the Herglotz type generalized variational principle is listed, and the Chaplygin equations for the nonconservative nonholonomic systems are deduced under the Suslov definition. In Subsection II B, the transformation of the invariance condition of the Herglotz principle is established. In Subsection II C, Noether’s theorem is established, and the Herglotz type conserved quantity is obtained. In Subsection II D, the inverse Noether theorem is established. In Sec. III, we take the Appell–Hamel problem with nonconservative forces as an example to illustrate the application of the results we obtained. Finally, the conclusion of this paper is drawn.
II. THEORETICAL PART
A. Herglotz type differential equations of motion
According to Ref. 14 , the Herglotz type generalized variational principle can be expressed as follows:
The functional z corresponds to the function , namely, . According to Ref. 14, the existence of functional z as a solution of partial differential Eq. (1) is easy to prove.
Equation (28) is the Herglotz type Chaplygin equation for the nonconservative nonholonomic system.
B. Condition for the invariance of the Herglotz principle
C. Noether’s conservation law of Herglotz type
By using formula (35), the Herglotz type Noether conservation law of nonconservative nonholonomic system can be obtained.
D. Inverse theorem of conservation law
III. EXAMPLE
IV. CONCLUSIONS
As the generalization of the classical Hamilton principle, the Herglotz generalized variational principle allows to more effectively study nonconservative systems. Different from previous studies under the Hölder definition in Ref. 30, this paper studied the conservation laws of Herglotz type for nonconservative nonholonomic systems based on the Suslov definition of the commutative relation between differential and variational operations. The main work of this paper is as follows:
First, the Herglotz type principle and Chaplygin equations were established under the Suslov definition, and the invariance condition of the principle was given.
Second, the conservation laws were established by using invariance conditions (Theorem 1).
Third, the inverse theorem of conservation laws for nonholonomic systems was given (Theorem 2).
The results and methods of this study can also be applied to Birkhoffian systems, fractional-order systems, and high-order nonholonomic systems.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12272248 and 11972241) and the Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant No. KYCX22_3251).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Xinchang Dong: Conceptualization (equal); Methodology (equal); Validation (lead); Writing – original draft (equal); Writing – review & editing (equal). Yi Zhang: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Methodology (equal); Project administration (equal); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.