We encounter many difficulties in describing many important physical progress with classical variational principles. For instance, the famous Hamilton principle for holonomic conservative systems, it encounters great difficulties in generalizing to non-conservative systems. In general, the Hamilton principle for holonomic non-conservative systems can be expressed as , and the formula can not be expressed in the form that the variation of an action functional is equal to zero, so it is no longer a stable action principle. For this purpose, Herglotz^[1] proposed a generalized variational principle and its action functional is defined by a differential equation. Different from classical variational principles, it can study non-conservative dynamical systems in a very good way. At the same time, it can handle the problems of conservative and non-conservative systems in a unified way. After that, Georgieva and Gueuther obtained the Noether theorem based on the Herglotz generalized variational principle.^[2,3] Torres et al. studied the higher-order Herglotz variational problem^[4] and Herglotz variational problem with time delay and its Noether theorem.^[5] Further, according to the Herglotz variational principle, Zhang studied the Noether’s theorem and conserved quantities in phase space,^[6] for non-conservative nonholonomic system,^[7] for Birkhoffian system,^[8–10] and with time delay.^[11,12] Recently, many results have been obtained about the fractional Herglotz variational principle.^[13–17]

The study of symmetry and invariants of dynamic systems is of great significance. Symmetry is a method to study the conservation of dynamical systems. There are mainly three different concepts of symmetry: Noether symmetry, Lie symmetry, and Mei symmetry. In addition, we can also study conserved quantities by using the differential variational principle. In 1986, Vujanovic studied the conservation laws of dynamical systems based on Jourdain’s principle and Gauss’s principle.^[18] After that, Liu studied the conservation laws of nonholonomic non-conservative dynamical systems based on Jourdain’s principle.^[19] Mei studied the conservation law of Birkhoffian systems by using Pfaff–Birkhoff–D’Alembert principle.^[20] Up to now, few papers on differential variational principle of Herglotz type have been published.

As we all know, an exact invariant can be found by studying conserved quantities. When a dynamic system is subjected to a small perturbation, the original conserved quantity will change. A physical quantity that almost does not change when a certain parameter of the system changes slowly with respect to the change of the parameter is called an adiabatic invariant. In 1917, Burgers^[21] first proposed the concept of adiabatic invariants through a special Hamiltonian system. After that, Beulanov,^[22] Kruskal,^[23] and Djukić^[24] obtained many important results. The research on the adiabatic invariants has aroused the interest of scholars and achieved some important results.^[25–36] To the author’s knowledge, the method of the differential variational principle of Herglotz type has not been applied to study the adiabatic invariant.

The layout of the article is as follows. We give the differential variational principle and exact invariant of Herglotz type in Section 2. In Section 3, we give a new type of adiabatic invariant for the non-conservative nonholonomic system. Besides, the existence conditions and form of the adiabatic invariant are presented. In Section 4, we give the inverse theorem of adiabatic invariant. In Section 5, we give an example to support our results. In Section 6, we give some conclusions to end the paper.

The differential variational principle of Herglotz type for the non-conservative nonholonomic system is^[7] 1 Here, is the Lagrangian of Herglotz type, and q_s(s = 1, 2, …, n) are the generalized coordinates of the system. The constraint equation is with s = 1, 2, …, n; β = 1, 2, …, g; ε = n – g; σ = 1, 2, …, ε.

Because of the independence of δq_σ, the differential equations of motion for this system are derived as follows: 2 If is the infinitesimal generator of space and f⁰ is the infinitesimal generator of time, then the isochronic variations are^[7] 3 where ε is an infinitesimal parameter.

If the infinitesimal generating functions and f⁰ satisfy the following conditions: 4 5 then there exists a conserved quantity for the non-conservative nonholonomic system of Herglotz type 6 where is a gauge function. If the system is not disturbed, the conserved quantity (6) is an invariant. We can call it an exact invariant.

Assume that the non-conservative nonholonomic system (2) is subjected to a small perturbation εQ_σ, that is, 7 Under the influence of εQ_σ, the primary symmetries and conserved quantities of the system will change accordingly. If the infinitesimal generating functions and of the system which are perturbed can be expressed as 8 and they satisfy 9 where G is a gauge function and satisfies 10 then we have the following theorem.

If the system is not subjected to nonholonomic constraints,the above theorem gives the corresponding results for holonomic non-conservative systems.

If the non-conservative nonholonomic system (2) is subjected to a small perturbation εQ_σ, then it has the following adiabatic invariant: 17 Because its orbit should satisfy formula (7), we have 18 Considering that 19 according to formula (14), and considering formulae (18) and (19), we can obtain 20 where f = f⁰ + εf¹ + ε²f² + ⋯ and . We can rewrite formula (20) as 21 So, if we have 22 assuming further 23 by Eqs. (22) and (23), we can find the generators of time f⁰ and space corresponding to the undisturbed part. Furthermore, we can also obtain 24 25 Hence, we have the following theorem.

Under the action of non-conservative force F_x = −x, F_y = −y, a unit mass particle moves along the horizontal plan xoy. The particle is constrained so that the slope of the trajectory is proportional to time t and the proportional coefficient is an unit. By means of the method expounded in this paper,^[7] we try to study the adiabatic invariants of the system.

The Lagrangian of the system of Herglotz type is 26 and the nonholonomic constraint is 27 There exists a conserved quantity 28 When the system is not disturbed, formula (28) is also an exact invariant.

Assume that the system is subjected to small perturbations 29 Equation (12) gives 30 Equation (30) has the following solutions: 31 By Theorem 1, we have 32 Formula (32) is the first order adiabatic invariant for the system. Similarly, we can obtain much higher order adiabatic invariants.

Lastly, we consider the inverse problem. If the system is disturbed by the small disturbance (29), then by formulae (22) and (23), we have 33 34 If we take 35 then we can obtain 36 Similarly, we have 37 38 If we take 39 then we can also obtain 40 Therefore, the generators (36) and (40) give the infinitesimal transformations corresponding to the first order adiabatic invariant (32).

It is a new way to study adiabatic invariants by using the differential variational problem of Herglotz type. The main work of the article is summarized as follows. (i) The exact invariant (6) of the non-conservative nonholonomic system and its existence conditions (4) and (5) are established. (ii) A new type of adiabatic invariant (13) for the non-conservative nonholonomic system is given and proved. (iii) The inverse problem of adiabatic invariants is discussed. The main results are two theorems and one corollary. The methods and conclusions can be further popularized and applied. For example, we can study the adiabatic invariant for a nonholonomic system on time scales based on Herglotz variational problem.