Abstract

This paper presents the Hamilton-Jacobi method for integrating the equations of motion of mechanical systems on time scales. We give the criterion and four basic forms of canonical transformation on time scales. Also, various examples are given to illustrate the role played by a generating function in the canonical transformation. By choosing an appropriate generating function, we construct the Hamilton-Jacobi equation on time scales and prove the Jacobi theorem on time scales. An example for an Emden-Fowler type equation is discussed to show the application of the method.

1. Introduction

As it is well known, the Hamilton-Jacobi equation [1–4], that is an important nonlinear partial differential equation, represents a reformulation of classical mechanics. In addition, the Hamilton-Jacobi method is very useful in integrating differential equations of motion for the holonomic mechanical systems [5–9], the nonholonomic mechanical systems [10, 11], and the nonconservative mechanical systems [12]. The differential equations that can be realized by Hamilton formularization [13–15] can also be solved by the Hamilton-Jacobi method [13, 14]. Moreover, the discrete analogues of the Hamilton-Jacobi equation have been studied by Elnatanov and Schiff [16] and Lall and West [17]. Ohsawa et al. [18] developed the discrete Hamilton-Jacobi theory within the framework of discrete Hamiltonian mechanics. Until now, the Hamilton-Jacobi theory is well understood both from the continuous and discrete points of view, and it has many important applications in optimal control problems, particle physics, fluid mechanics, quantum mechanics, and cosmology. In this paper, the Hamilton-Jacobi method for mechanical systems is studied in version of time scales.

A time scale is an arbitrary nonempty closed subset of the real numbers. The calculus of time scales, which has recently attracted a lot of attention, was introduced by Hilger [19] in 1988 to unify continuous and discrete analysis. In terms of a unique formalism, this theory can deal with not only continuous and discrete analysis but also complex processes such as the control systems, impulsive dynamical systems, electromechanical systems, neural network, and economical systems [20–24]. The calculus of variations on time scales that was one of the popular topics has been well studied [25–35]. Inspired by these works, the Euler-Lagrange equations, the Hamilton canonical equations [36–38], and the Birkhoff’s equations [39] were established for mechanical systems on time scales. Besides, the Noether’s theorems [37–42] were established in finding conserved quantities for mechanical systems on time scales. However, it is worth mentioning that the famous Hamilton-Jacobi method also represents an important integration method. It is worth and necessary to study the Hamilton-Jacobi method on time scales.

The outline of this paper is as follows: we first present the basic definitions and properties about the calculus of time scales in Section 2. In Section 3, we give the Hamilton canonical equations on time scales. Section 4 focuses on the canonical transformation theory on time scales. In Section 5, we construct the Hamilton-Jacobi equations on time scales and prove the Jacobi theorem on time scales. Section 6 gives the conclusions and future directions of research.

2. Preliminaries on Time Scales

We begin by introducing the definitions and properties on time scales needed in the sequel. More discussions and proofs can be found in [21, 24].

A time scale is an arbitrary nonempty closed subset of the real numbers . For all , the forward jump operator is defined by , while the backward jump operator is defined by , with and .

A point is called right-dense, right-scattered, left-dense, and left-scattered if , , and , respectively. The graininess function is defined by . If has a left-scattered maximum , then we define , otherwise .

Let be a function. Then the delta derivative of at a point is defined to be the number with the poverty that given any , there exists a neighborhood of (i.e., ) for some such that

A function is called rd-continuous if it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of rd-continuous functions can be denoted by . The set of differentiable functions with rd-continuous derivative is denoted by .

A function is called an antiderivative of provided holds for all . Then the indefinite integral of is defined by , where is an arbitrary constant. The definite integral is defined by , for all .

For delta differentiable and , the next formulae hold where we abbreviate by , that is, .

Let be a function. The partial delta derivative of respect to is defined as the limit

Let the function be -completely delta differentiable at the point . If the functions and have delta derivatives at the point , then the composite function has a delta derivative at that point which is expressed by the formula where , .

Remark 1. If and , then (8) means that

Remark 2. If the function be -completely delta differentiable at the point , if the functions and have delta derivatives at the point , then the composite function has a delta derivative at that point which is expressed by the formula where , .
The formulae (8) and (11) constitute a dual pair in which the shift appears at two different places. This is a natural phenomenon in the calculus on time scales such as the product rule (3) has two forms.

Let a function have the mixed delta derivatives and in some neighborhood of the point . If these derivatives are continuous at the point , then

3. Hamilton Canonical Equations on Time Scales

In this section, we present the Hamilton canonical equations on time scales that have been discussed in literatures [36–38].

Assuming that the configuration of a mechanical system is determined by generalized coordinate , the Lagrangian is , where , , . The Hamilton action is

The Hamilton principle on time scales can be expressed as which satisfies the relationships and the boundary conditions

The generalized momentum and the Hamiltonian are

The Hamilton canonical equations on time scales can be derived from the principle above, that is,

4. Canonical Transformations on Time Scales

Now, we consider the transformation from the old variables , to the new variables , and suppose that (19) is invertible. Undergoing the transformation, the canonical equations (18) on time scales will be transformed into other new equations. However, the new equations may not have the canonical form in general. Among the possible transformations, there is a class of transformations called canonical transformations whose properties are particularly useful in dynamics. That is, the canonical equations (18) are transformed into the new canonical equations where the new Hamiltonian is

The following criterion gives the condition under which (19) is canonical on time scales.

Criterion 1. If (19) satisfies the condition then it is a canonical transformation on time scales, where is delta differentiable.

Proof 1. Formula (22) can be written as suppose that and exist with the isochronal variations and . Then (23) becomes Taking the delta derivative of (24) with respect to , we obtain Taking the variation of (22), we have By using (15) and subtracting (25) from (26) yields According to (3), (27) becomes Taking note of (18), we obtain Therefore, (20) can be derived from (29).

In fact, (22) is also the necessary condition of the canonical transformation on time scales. According to the Hamilton principle on time scales, if both the new variables and old variables satisfy then (19) is canonical on time scales. From (30) and (31), the two integrands should satisfy the following relation where can be an arbitrary function with new variables, old variables, and time.

Because the canonical transformation completely depends on the choice of the arbitrary function , the function can be called the generating function. Clearly, and can be solved from (23), and the solution of the canonical transformation problem on time scales is not unique. In order to achieve the transformation between the two groups of canonical variables, the generating function should contain both new variables and old variables [1]. With that case in mind, the four forms of the canonical transformations on time scales are discussed below, and some simple and important examples are given to illustrate the effect of the canonical transformation on time scales and the generating function.

Case 1. The generating function of the first kind has the form we have Substituting (34) into (23), we obtain From (35), we can find

Example 1. If the generating function has the form then the corresponding canonical transformation gives The transformation (38) shows that the new generalized coordinates depend on the old generalized momentum while the new generalized momentum depends on the old generalized coordinates.

Case 2. The generating function of the second kind has the form combining (39) and (23), we have From (40), we obtain

Example 2. If the generating function has the form then the corresponding canonical transformation gives

Case 3. The generating function of the third kind has the form Similarly, we have

Example 3. If the generating function has the form then the corresponding canonical transformation gives The transformation (47) shows that the new generalized coordinates and the new generalized momentum are the same with the old ones. Hence, the generating function (46) is corresponding to the identical transformation.

Case 4. The generating function of the fourth kind has the form Similarly, we have

Example 4. If the generating function has the form then the corresponding canonical transformation gives Compared with (38), (51) is actually the same with (38). Hence, different generating functions can correspond to the same canonical transformation.

The canonical transformations expressed by the four kinds of generating functions above are only part of the canonical transformations on time scales. However, it is quite extensive. If is not explicitly contained in , , , and , then , (23) can be written as This kind of transformation can be called contact transformation on time scales.

5. Hamilton-Jacobi Method on Time Scales

The form of Hamiltonian will be simpler if we choose an appropriate generating function, such as . If , (20) becomes

By integrating, we have where and are constants and , .

In order to achieve the purpose above depends on the generating function we choose. According to the relationship between the new Hamiltonian and the old Hamiltonian Thus, the generating function should satisfy

If we choose and use symbol take the place of , then we can obtain the following equation

Substituting into , we have

Obviously, can be determined by initial conditions. Hence, the transformation relation corresponding to the function can be written as

Substituting into (57), we obtain

Equation (60) can be called the Hamilton-Jacobi equation on time scales.

Theorem 1. If function is a complete integral of the Hamilton-Jacobi equation (60) on time scales, that is, , is the constant of integration, and and it satisfies the Hamilton-Jacobi equation (60) on time scales, then the first integrals of the canonical equations (18) on time scales are determined by

Proof 2. First, we show that (62) is the first integral of the system on time scales, namely, Taking note of (18) and (64), we have Taking the partial delta derivative of (60) with respect to and noting (9), we have Notice that (66) coincides with (65). Therefore, (64) is an identity.
Similarly, we verify that (63) is also the first integral. Actually, taking the delta derivative of (63) with respect to , we have Then (67) becomes Taking the partial delta derivative of (60) with respect to yields Considering and (18), we know that (69) coincides with (68), that is, (68) is also an identity, completing the proof.