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.

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.

According to Ref. 14 , the Herglotz type generalized variational principle can be expressed as follows:

Let the functional z be determined by a first-order differential equation,
(1)
Try to determine the trajectory qst such that zb achieves an extreme value, i.e.,
(2)
with the initial condition
(3)
where a, b, zaR, Lt,qs,q̇s,z is the Lagrangian of Herglotz type, qs (s=1,2,…,n) is the generalized coordinate and satisfies the boundary conditions qstt=a=qsa,qstt=b=qsb, and q˙ss=1,2,,n is the generalized velocity.

The functional z corresponds to the function qst, namely, z=zqs;t. According to Ref. 14, the existence of functional z as a solution of partial differential Eq. (1) is easy to prove.

If the motion of the mechanical system is restricted by g nonholonomic constraints, i.e.,
(4)
According to the Appell–Cheatev condition,12 the restriction of the nonholonomic constraints (4) exerted on the virtual displacements is
(5)
According to the Suslov definition of the reciprocity relation between differential and variational operations, we have the following exchange rules:12 
(6)
and
(7)
where
(8)
Taking isochronous variation on both sides of Eq. (1), we get
(9)
On the one hand,
(10)
while on the other hand,
(11)
By subtracting formula (11) from formula (10), we obtain
(12)
According to the formula
(13)
we have
(14)
i.e.,
(15)
Substituting formulas (7) and (15) into (12), we can get
(16)
Substituting formula (9) into (16), we can get
(17)
In Eq. (17), let Lqsδqs+Lq̇sδq̇s+Lq̇ε+βTσε+βδqσ=A; then it can be rewritten as
(18)
Equation (18) can be regarded as a first-order ordinary differential equation with respect to the variable δz. According to the existence and uniqueness theorem of the solution,33 the solution of Eq. (18) exists as follows:
(19)
Using the initial conditions (3) and taking into account Eq. (2), from Eq. (19), we immediately obtain
(20)
Let L̃qs,q̇σ,t,z be the Lagrangian L for eliminating the non-independent generalized velocity q̇ε+β by using constraint (4). Similar to Ref. 13, it is easy to derive the following relations:
(21)
and
(22)
Multiplying both sides of formula (21) by δqs, we get
(23)
Using condition (5) satisfied by the virtual displacement, from formula (23), we can get
(24)
From formula (22), and considering condition (5), we obtain
(25)
By substituting formulas (24) and (25) into Eq. (20), integrating by parts, and considering the arbitrariness of the integration interval, we can get
(26)
Equation (26) is the Herglotz type differential variational principle of the system. Due to the independence of δqσ, we have
(27)
Equation (27) can also be written as
(28)

Equation (28) is the Herglotz type Chaplygin equation for the nonconservative nonholonomic system.

Introducing space and time generators ξs and ξ0, we have
(29)
where ɛ is an infinitesimal parameter. According to the relation between isochronism and non-isochronism variation,
(30)
we can get
(31)
Substituting formula (31) into condition (5), we have
(32)
Equation (32) is the restriction of nonholonomic constraint (4) on generators, also known as the Appell–Cheatev condition.34 By substituting formula (31) into principle (26), we obtain
(33)
By adding and subtracting εddtGexpatLzdθ in (33), where G=Gt,qs,q̇σ,z is gauge function, and taking note of
(34)
we obtain
(35)
Equation (35) is the transformation of the invariance condition of principle (26).

By using formula (35), the Herglotz type Noether conservation law of nonconservative nonholonomic system can be obtained.

Theorem 1.
For the nonconservative nonholonomic system determined by (28) and (4), if the infinitesimal generators ξσ and ξ0 and the gauge function G satisfy the structural equation
(36)
and the Appell–Cheatev condition (32), then the conserved quantity of Herglotz type exists, i.e.,
(37)

Proof.
Substituting Eq. (36) into transformation (35), we can obtain
(38)
By integrating it, the conserved quantity (37) is obtained. The theorem is proved.□

Suppose that system (27) has a conserved quantity,
(39)
Taking the derivative of (39), we have
(40)
Multiplying Eq. (27) by ξσq̇σξ0, and taking the sum of σ, we have
(41)
In comparison with the coefficient of q̈σ in formulas (40) and (41), one has
(42)
Let the given conserved quantity be equal to (37), i.e.,
(43)
The generators ξσ, ξ0, and G can be solved from Eqs. (42) and (43). The following theorem may be obtained.

Theorem 2.

If there has a conserved quantity in the form of formula (37) for system (27), then the generators ξσ and ξ0 and gauge function G can be found from Eqs. (42) and (43).

To study the Appell–Hamel problem,11,35 assume that the system is also subject to nonconservative forces such as
(44)
Thus, the Herglotz type Lagrangian is
(45)
According to Eq. (1), we have
(46)
This is the equation that the functional z of this problem needs to satisfy.
The nonholonomic constraint equation of this system is35 
(47)
By using Eq. (47), we can eliminate q̇3 in formula (45) and obtain
(48)
According to Eq. (28), we have
(49)
The Appell–Cheatev condition (32) gives
(50)
and the structural Eq. (36) gives
(51)
Equations (50) and (51) have the following solutions:
(52)
(53)
(54)
By using Theorem 1, we obtain the following conserved quantities, respectively:
(55)
(56)
(57)
Now we study the inverse problem. Suppose there is a conserved quantity
(58)
According to formulas (42) and (43), we have
(59)
(60)

There are five unknowns in Eqs. (59) and (60), so their solutions are not unique.

If ξ1 = 0, then we get
(61)
If ξ1=2q̇3q̇1et, then we get
(62)

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.

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).

The authors have no conflicts to disclose.

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 sharing is not applicable to this article as no new data were created or analyzed in this study.

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