Time scale is an arbitrary nonempty closed subset of the set of real numbers. It unifies and extends the theories of continuous and discrete cases. ^[1–6] The theories of calculus and dynamic equations on time scales are applied to many different fields, such as physics, economics, optimal control, and so on. The theories of time scales was introduced by Hilger. ^[7] In 1988, Hilger ^{[8, 9]} studied the calculus on time scales, in which the difference and differential equations were unified and extended.

In 2001, the book about dynamic equations on time scales was published by Bohner and Peterson, ^[10] the definitions and propositions of calculus on time scales were introduced, and the first-order linear equations and the second-order linear equations were established on time scales, respectively. The calculus of variations on time scales was initiated by Bohner using the delta derivative and delta integral. ^[11] The Euler–Lagrange equations of a first-order variational problem with delta derivative were derived. Atici, Biles, and Lebedinsky ^[4] discussed the theories of integral on time scales, which have been applied in the fields of economics. The Euler–Lagrange equations were obtained for a first-order variational problem on time scales involving nabla derivatives instead of delta ones. Martins and Torres ^[12] extended the variational problem with delta derivative to the case with nabla derivative, and the necessary optimality condition of the Euler–Lagrange equations for higher order variational problem with nabla derivative was proved by using a new and more general fundamental lemma of the calculus of variations on time scales. Later, Torres ^[13] further discussed the variational problems both with delta and nabla derivatives, in which the variational problems of the discrete, the quantum, and the continuous were unified and extended with delta and nabla derivatives.

In 2008, Bartosiewicz and Torres ^[14] studied the Noether theorem of Lagrange systems on time scales. The Noether theorem of the systems on time scales were given based on the invariance with transforming time and without transforming time. Martins and Torres ^[15] discussed the Noether symmetry theorem of variational problems with nabla derivatives, they proved a Noether-type symmetry theorem and a Dubois–Reymond necessary optimality condition for the calculus of variational with nabla derivative. Malinowska and Ammi ^[16] extended the Noeher theorem on time scale to control problems, and the Noether theorem for optimal control problems defined on delta derivative was proved. Meanwhile, the Noether symmetries of nonholonomic systems on time scales were discussed by Fu and his coworkers. ^[17] Zhang and Song ^[18] extended the Noether symmetries of mechanical systems to Birkhoffian systems on time scales. Some results of the variational problems and Noether symmetries of mechanical systems on time scales have been obtained, ^[19–22] while the methods of reduction for the systems on time scales have not been investigated yet in the literature.

While for the holonomic mechanical systems, the Routh function is constructed to eliminate the circlic coordinates by using the the cyclic integrals, and the Whittaker function is constructed to decrease the number of generalized coordinates by using the generalized energy integrals. The equations of motion for the mechanical systems are reduced, the reduced equations are still hold the form of equations of motion for the systems, it is the famous Routh ^[23] and Whittaker ^[24] methods of reduction in mechanics systems. In the past twenty years, the Routh and Whittaker methods of reduction in mechanics systems have been extended, and a series of important research results have been obtained. ^[25–30] Up to mow, the research of the methods for order reduction is limited to the standard mechanical systems, but it has not been related to the mechanical systems on time scales.

In this paper, we mainly study the Routh and Whittaker reduction methods by using the cyclic integrals and generalized energy integrals of Lagrange systems on the time scale. The structure of this paper is as follows. In Section 2, the definitions and propositions on time scales are given. The cyclic integral of Lagrange system with nabla derivative is defined in Section 3. The Routh method of reduction for Lagrange system with nabla derivatives is derived in Section 4. In Section 5, the generalized energy integral of Lagrange system with nabla derivative is derived. The Whittaker method of reduction for Lagrange system with nabla derivative is studied in Section 6. In Section 7, two examples are given to illustrate the application of the results.

In this section, we review some basic definitions and properties of calculus on time scales used in the following sections briefly. Detailed discussions and proofs can be found in Ref. [10].

A time scale is a nonempty closed subset of the set of real numbers . Let be an arbitrary time scale, their denote is the forward jump operator

(1)

(2)

A point is said to be right dense, right scattered, left dense, right scatter, if , and , respectively. A point t is isolated if , and t is dense if . The (backward) graininess function is defined by

(3)

In order to introduce the definition of a nabla derivative, a set is defined, which is derived from as follows: if has a right scattered minimum m, then , otherwise .

Let function , then for all , is defined by

(4)

Let function , if for all , there exists a neighborhood of t (i.e., ), such that

(5)

A function is called ld-continuous if it is continuous at the left dense points and its right sided limits exists (finite) at all right dense points in . The set of all ld-continuous functions is denoted by , and the set of all nabla differentiable functions with ld-continuous derivative given by .

A function is called a nabla antiderivative of , if for all , provided . In this case the nabla integral of f from a to is defined by .

The propositions of calculus on time scales are as follows:

(6)

(7)

(8)

Assume that the configuration of a mechanical system is determined by n generalized coordinates . The Lagrangian is , and the Hamilton principle of the system with nabla derivative is

(9)

(10)

(11)

Formula (9) can be expressed as

(12)

Considering formula (6) and the conditions (11), we have

(13)

Substituting formula (13) into formula (12), we have

(14)

From Lemma 1, we have

(15)

(16)

If some of coordinates are not contained in Lagrangian L, for instance , i.e.,

(17)

(18)

Integrating Eq. (18), we have

(19)

For the Lagrange system with nabla derivatives, the equations of motion for the system are reduced by using the cyclic coordinates. In this section, we establish the Routh method of reduction for Lagrange system with nabla derivative.

Suppose that the k coordinates of the Lagrange system with nabla derivative are not explicit, then the system is the k cyclic integrals as

(20)

(21)

Define the Routh function with nabla derivative as

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

In fact

If then equations (28) are reduced to the Routh equations of classical Lagrange system as

(30)

If , is the discrete Routh function, then equations (28) are reduced to Routh equations of discrete Lagrange system as

(31)

Hence, a dynamical problem on time scales with n degrees of freedom, which has k cyclic coordinates, can be reduced to a dynamical problem on time scales which only has n–k degrees of freedom.

In order to get the corresponding generalized energy integral of Lagrange system with nabla derivative, we calculate

(32)

From Eqs. (16) and (32), if the time t is not explicit in function L, and formula (32) satisfies the condition

(33)

(34)

(35)

Choose a generalized coordinate to substitute the effect of the time t, such as q ₁. Let

(36)

(37)

Suppose that

(38)

Differentiating formula (38) with respect to , and , we obtain respectively

(39)

(40)

(41)

Using formula (35), we can obtain as

(42)

From formula (38) and (39), formula (35) can be expressed as

(43)

(44)

(45)

Construct the Whittaker function with nabla derivative as

(46)

Differentiating formula (46) with respect to and , we obtain

(47)

(48)

Comparing Eq. (44) with Eq. (47), we obtain

(49)

Comparing Eq. (45) with Eq. (48), we obtain

(50)

According to Eqs. (40) and (41), we obtain

(51)

(52)

(53)

(54)

If , then equations (53) or (54) can be written as the Whittaker equations of classical Lagrange system ^{[33, 34]}

(55)

(56)

If , and is the discrete Whittaker function, then equations (53) or (54) can be written to Whittaker equations of the discrete Lagrange system as

(57)

(58)

Hence, the integral of energy enables us to reduce a given dynamical system with nabla derivative with n degrees of freedom to another dynamical system with nabla derivative with degrees of freedom.

Clearly, is a cyclic coordinate with nabla derivative and the integral corresponding to the cyclic coordinate of the system is

(60)

The Routh function is

(61)

From formula (60), we obtain

(62)

Substituting formula (62) into formula (61), we obtain

(63)

Substituting Eq. (63) into Eq. (28), we obtain

(64)

From formula (29), we obtain

(65)

If formulae (64) and (65) can be written as

(66)

(67)

If , formulae (64) and (65) are expressed as the Routh equation of discrete Lagrange system

(68)

(69)

From formula (35), we obtain

(71)

Satisfy the condition

(72)

Choose a generalized coordinate q ₁ to substitute the effect of the time t. Setting

(73)

(74)

Substituting formula (74) into (71), we have

(75)

Therefore, we have

(76)

Construct the function Ω as

(77)

The Whittaker function of the system is

(78)

Substituting formula (78) into equation (53) or (54), we obtain

(79)

(80)

Since is not contained in W, we get a new integral of energy

(81)

Therefore, we have

(82)

From formula (9), substituting formula (82) into formula (74), and integrating it, we obtain

(83)

If , then formulae (75) and (78) can be expressed as

(84)

(85)

The Whittaker equation (79) or (80) is written as

(86)

(87)

Formulae (82) and (83) are expressed as

(88)

(89)

If , then formulae (75) and (78) can be expressed as

(90)

(91)

Equation (80) or (81) is written as

(92)

(93)

The theories of calculus for continuous, discrete, and quantum cases, have been unified and extended by the theories of calculus on time scales. Considering the complexity of dynamical behavior of mechanical systems on time scales, the research of integral theories for mechanical systems on time scales is still in the initial stage of development. In this paper, we study the methods of reduction for Lagrange system with nabla derivative. The Lagrange equations for the system are established, and their cyclic integrals and generalized energy integrals are given. The order of Lagrange equations (16) is reduced, the Routh equations (28) are obtained by using the cyclic integrals, and the Whittaker equations (53) or (54) are derived by using the generalized energy integrals with nabla derivatives. The results show that the reduced Routh equations and Whittaker equations hold the form of Lagrange equations with nabla derivatives. Comparing with the Lagrange equations of the system, the Routh equations (28) have only n–k equations and the Whittaker equations (53) or (54) have only equations, and the cases of continuous and discrete are two special cases with nabla derivative. The results of this paper are of universal significance, the theories of mechanical systems on time scales can be further generalized to the theories of Birkhoffian system, optimal control, and so on.