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Mathematical Problems in Engineering
Volume 2012, Article ID 345942, 34 pages
http://dx.doi.org/10.1155/2012/345942
Research Article

Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part II: Foliation Structures and Integrating Algorithms

Department of Applied Electronics, Faculty of Industrial Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan

Received 6 May 2012; Revised 3 July 2012; Accepted 4 July 2012

Academic Editor: Wei-Chiang Hong

Copyright © 2012 Tatsuya Kai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates foliation structures of configuration manifolds and develops integrating algorithms for a class of constraints that contain the time variable, called A-rheonomous affine constrains. We first present some preliminaries on the A-rheonomous affine constrains. Next, theoretical analysis on foliation structures of configuration manifolds is done for the respective three cases where the A-rheonomous affine constrains are completely integrable, partially integrable, and completely nonintegrable. We then propose two types of integrating algorithms in order to calculate independent first integrals for completely integrable and partially integrable A-rheonomous affine constrains. Finally, a physical example is illustrated in order to verify the availability of our new results.

1. Introduction

Recent years, interest on nonholonomic systems [15] has been higher in the various research fields such as analytical mechanics, control theory, robotics, among others. Nonholonomic systems are defined as systems which are subject to some nonintegrable constraints and behaviors of the systems have to satisfy the constraints. Various examples of nonholonomic systems can be easily found, for example, mobile cars [6, 7], trailers [8, 9], space robots [10, 11], acrobat robots [12, 13], a rolling ball or coin on a plain [1], underactuated manipulators [1416], among others. In a lot of researches on nonholonomic control systems, linear constraints, which are represented in the form: , , , have been mainly considered so far. The class of the linear constraints contains many examples such as mobile cars, a ball or a coin on a plain, hopping robots [14, 15], among others. It is well known that integrability and nonintegrability of the linear constraints can be determined by using well-known Frobenius’ theorem [17, 18], and there is a strong relationship between them and accessibility of nonlinear control systems with the linear constraints [4, 5, 7, 1721]. Moreover, for kinematic and dynamic systems subject to the linear constraints, the following results are also well known: (i) their linear approximated systems are uncontrollable, (ii) they are locally controllable, but not locally asymptotically stabilizable by any nonlinear smooth state feedback laws from Brockett’s theorem [4, 22]. Hence, a lot of work have been done to overcome the difficulties until now.

However, the class of the linear constraints does not cover all of the nonholonomic systems. There exists a larger class of constraints and it is called scleronomous affine constraints, which are represented as , , , . The terminology “scleronomous” means “time-invariant” and is mainly used for constraints. A space robots with an initial angular momentum (Figure 1(a)), a boat or a ship on a running river (Figure 1(b)), a ball or a coin on a rotating table (Figures 1(c) and 1(d)) are typical examples of systems subject to the scleronomous affine constraints. Compared to researches on the linear constraints, ones on the scleronomous affine constraints are quite few, but we have focused on the scleronomous affine constraints and derived various results from the viewpoints of both mathematics and nonlinear control theory [2332]. In work on the scleronomous affine constraints, the following new results, which are beyond the results on the linear constraints mentioned above, have been obtained: (a) complete nonintegrability (nonholonomicity) of the scleronomous affine constraints is equivalent to strong accessibility of systems, (b) linear approximated systems of original systems with scleronomous affine constraints are controllable under some conditions, (c) some systems with scleronomous affine constraints satisfy Brockett’s condition and hence there is a possibility of local asymptotic stabilizability by nonlinear smooth state feedback laws. Furthermore, we can consider a larger class of constraints which contains the time variable, that is to say, rheonomous constraints, and a boat or a ship on a running river with a time-varying stream (Figure 1(b)) and a coin or a ball on a rotating table at a time-varying angular velocity (Figures 1(c) and 1(d)) are expected to be subject to rheonomous constraints. Since rheonomous constraints contain the time variable explicitly, it is more difficult to analyze and control systems subject to them in comparison with the scleronomous constraints case. To make matters worse, the results on scleronomous constraints cannot be utilized for rheonomous constraints, hence, we need fundamental theory for rheonomous constraints to consider analysis and control of systems with them. As a matter of course, such rheonomous constraints have been already treated in [3337] by using various mathematical tools, however, fundamental properties for them such as integrability/nonintegrability and foliation structures of configuration manifolds have not been derived in these work.

fig1
Figure 1: Examples of systems subject to affine constraints.

This is the second paper of a series of papers that deal with a class of rheonomous constraints, called -rheonomous affine constraints defined on configuration manifolds. The -rheonomous affine constraints are represented in the form , that is to say, affine constraints whose term contains the time variable . The class of the -rheonomous affine constraints is larger than the scleronomous affine constraints and contains some physical and mechanical examples as shown in Figure 2. In the first paper [38], we show some fundamental properties of the -rheonomous affine constraints and derive some necessary and sufficient conditions on integrability/nonintegrability for them. On the other hand, in this second paper, we will focus on analysis of foliation structures of configuration manifolds with the -rheonomous affine constraints, and development of integrating algorithms to calculate independent first integrals for integrable -rheonomous affine constraints.

345942.fig.002
Figure 2: Classes of constraints and examples.

The outline of this paper is as follows. First, we give some preliminaries on the -rheonomous affine constraints in Section 2. In Section 3, we next investigate the relationships between the -rheonomous affine constraints and foliation structures of configuration manifolds in the respective three cases: complete integrability, partial integrability, and complete nonintegrability. Then, in Section 4, we develop two types of integrating algorithms to calculate independent first integrals for completely integrable and partially integrable -rheonomous affine constraints. Moreover, theoretical analysis of the algorithms is also shown. Finally, we illustrate a physical example, a pendulum with a time-varying elastic string, in order to verify the availability of our new results in Section 5. Throughout this paper, manifolds, submanifolds, functions, vector fields, distributions, and differential forms are all assumed to be smooth.

2. Preliminaries

This section presents some preliminaries on rheonomous affine constraints which we deal with throughout this paper. See also the first paper [38] for more details. First, we define rheonomous affine constraints and explain geometric representation of them. We denote the time variable by and a time interval by . Let be an -dimensional configuration manifold and be a local coordinate of . Associated with , we refer as a tangent vector field.

Throughout this paper, we deal with -rheonomous affine constraints: Note that the coefficients in (2.1) depend on the time variable . We rewrite (2.1) as where a rheonomous affine term is a vector-valued function whose th entry is , and is a matrix-valued function whose th entry is . It must be noted that the class of the -rheonomous affine constraints contains some important examples of mechanical systems as mentioned in Section 1. We here assume a sufficient condition on independency of the -rheonomous affine constraints (2.2) as follows.

Assumption 2.1. The coefficient matrix of the -rheonomous affine constraints (2.2) has a row full-rank at any point , that is, holds.

Now, we consider a classification method of all the points in the configuration manifold in terms of the rheonomous affine term in the -rheonomous affine constraints (2.2) by the following definition.

Definition 2.2 (see [38]). For the rheonomous affine term of the -rheonomous affine constraints (2.2) and a time interval , a point such that , holds is called a rheonomous affine regular point. On the other hand, a point such that , holds is called a rheonomous affine equilibrium point.

Note that from Definition 2.2, each point in is classified into either a rheonomous affine regular point or a rheonomous affine equilibrium point. Next, we explain geometric representation and some fundamental properties for the -rheonomous affine constraints, which are originally obtained in the first paper [38]. From (2.3) in Assumption 2.1, the row vectors of in the -rheonomous affine constraints (2.2) are independent of each other at any point . So, we consider vectors which are independent of each other and annihilators of the row vectors of , and denote them by as vector fields on . Moreover, we also denote a space spanned by , that is, a distribution on by Since the basial vectors of : are independent of each other at any point , is a nonsingular distribution, that is, holds. A curve is said to be satisfied the -rheonomous affine constraints (2.2) if for a vector field on : and the generalized velocity of : : holds as shown in Figure 3. We call this a rheonomous affine vector field.

345942.fig.003
Figure 3: Geometric representation of -rheonomous affine constraints.

We then show an important characteristic of the rheonomous affine vector field as the next proposition.

Proposition 2.3 (see [38]). For the -rheonomous affine constraints (2.2), the component of the rheonomous affine vector field , and a time interval : holds.

Under the preliminaries shown above, we define the geometric representation method for the -rheonomous affine constraints as the following definition.

Definition 2.4 (see [38]). The -rheonomous affine constraints (2.2) are geometrically represented by a pair , where is an -dimensional distribution defined by (2.4) and is called a rheonomous affine vector field and satisfies (2.7).

In order to derive some theoretical analysis on -rheonomous affine constraints (2.2), the geometric representation of them is quite important. For the geometric representation of the -rheonomous affine constraints shown in Definition 2.4, the next proposition can be derived.

Proposition 2.5 (see [38]). For the geometric representation of the -rheonomous affine constraints (2.2): and a time interval , a point is a rheonomous equilibrium point if and only if holds. On the other hand, a point is a rheonomous regular point if and only if holds.

Finally, we explain the rheonomous bracket which is a new operator and originally introduced in the first paper [38]. Similar to the first paper [38], the rheonomous bracket also plays important roles in this second paper. The Lie bracket of two vector fields is defined as an operator defined on : Based on the Lie bracket above, the rheonomous bracket is defined as follows.

Definition 2.6 (see [38]). For the vector fields defined on of the geometric representation of the -rheonomous affine constraints (2.2): , the rheonomous bracket is an operator: : that satisfies the next three properties.(a)For a rheonomous affine vector field , holds.(b) is defined as a set of vector fields that consists of and iterated rheonomous brackets of and does not contain . For a rheonomous affine vector field and a vector field , hold.(c)For two vector fields , hold.

Note that, in Definition 2.6, the rheonomous affine vector field is perceived as special, and this yields an additional term for time differential of a vector field as shown in the property (b). It is also noted that the rheonomous bracket is equivalent to the normal Lie bracket for scleronomous affine constraints, that is, constraints that do not contain the time variable explicitly. Moreover, the following proposition indicates that the rheonomous bracket has the important characteristics in common with the normal Lie bracket.

Proposition 2.7 (see [38]). For the vector fields on the geometric representation of the -rheonomous affine constraints (2.2): and the set of iterated vector fields of them: , the following properties (a), (b), and (c) hold.(a)Bilinearlity: (b)Skew-symmetry: (c)Jacobi’s identity:

3. Foliation Structures of Configuration Manifolds

3.1. Completely Integrable Case

In this section, we investigate foliation structures of configuration manifolds for the respective three cases where the -rheonomous affine constraints are completely integrable, partially integrable, and completely nonintegrable. It is quite essential to comprehend foliation structures of configuration manifolds in terms of model reduction theory, accessibility analysis of nonholonomic control systems, and so on.

First, we consider the complete integrability case. We now define a smallest and involutive time-varying distribution which contains and iterated rheonomous brackets of them, and satisfies , for all , that is to say, is spanned by all the rheonomous brackets of with the exception of . In addition, we also define a smallest and involutive time-varying distribution which contains all the iterated rheonomous brackets of . We here show necessary and sufficient conditions on complete integrability for the -rheonomous affine constraints (2.2), which are originally derived in the first paper [38].

Theorem 3.1 (see [38]). For the -rheonomous affine constraints (2.2) defined on an -dimensional manifold and a time interval , the following statements (a)–(c) are equivalent to each other. If they hold, the -rheonomous affine constraints (2.2) are said to be completely integrable or completely holonomic.(a)There exist independent first integrals of the -rheonomous affine constraints (2.2): such that hold, where is the exterior differential operator defined on .(b)There exists an -dimensional time-varying integral manifold of the -rheonomous affine constraints (2.2) determined by an initial point such that holds.(c)For a smallest and involutive time-varying distribution , holds.

From the statement (b) in Theorem 3.1, we can see that there exists an -dimensional time-varying integral manifold in the -dimensional configuration manifold . It is expected that this time-varying integral manifold creates a foliation structure in . Hence, we investigate the foliation structure with much circumstance. Now, we show the following lemma, which is useful for the proof of the main theorem.

Lemma 3.2. Assume that the -rheonomous affine constraints (2.2) are completely integrable. Onedenotes a set of rheonomous affine regular points by and a set of rheonomous affine equilibrium points by where is a time interval. Then, the following statements (a) and (b) hold.(a)For and , hold, where denotes the empty set.(b)For a neighborhood of a rheonomous affine equilibrium point , holds.

Proof. Since each point in is classified into either a rheonomous affine regular point or a rheonomous affine equilibrium point from Definition 2.2, the statement (a) is determinately satisfied. Next, since a rheonomous affine equilibrium point satisfies from Proposition 2.5, we can represent as with functions . Consider vector fields which are independent of each other and are orthogonal to . Calculating inner products of and , we have where is the inner product operator of two vector fields. Therefore, since (3.9) means equations for the point , it can be confirmed that the dimension of a neighborhood of : is greater than or equal to from the implicit function theorem [39, 40]. Consequently, the proof of this lemma is completed.

Now, we derive the main results on a foliation structure of the configuration manifold for the completely integrable -rheonomous affine constraints.

Theorem 3.3. Assume that the -rheonomous affine constraints (2.2) are completely integrable. Then, the following statements (a) and (b) hold for a time interval .(a)For a neighborhood of a rheonomous regular point : , there exist a local coordinate and an -dimensional time-varying foliation structure in which is constructed by an -dimensional time-varying integral manifold determined by the time variable and the point : where is a flow (one-parameter local transformation group) of . In addition, hold.(b)For a neighborhood of a rheonomous equilibrium point : , there exist a local coordinate and an -dimensional time-invariant foliation structure in which is constructed by an -dimensional time-invariant integral manifold determined by the point : In addition, hold.

Proof. First, we prove the statement (a). See also the illustration depicted in Figure 4 and the proof of Theorem 3.1 in the first paper [38] for understanding the proof. Let us denote the -dimensional expanded configuration manifold which is constructed by the product manifold of the space of the time variable and the configuration manifold by with the coordinate . The -rheonomous affine constraints (2.2) defined on can be represented by an expanded Pfaffian equation on : with expanded differential forms: We can see that annihilators of the expanded differential forms (3.17) as We also consider a diffeomorphism which is a projection from to at the time : , and its derivation: . Since holds from the definition of in(3.18), we obtain . In addition, from , it turns out that . Thus, by using a local coordinate , we can represent the -dimensional time-varying integral manifold for the -rheonomous affine constraints (2.2): as (3.10). Furthermore, from the definitions of and , it can be confirmed that (3.11) and (3.12) holds.

345942.fig.004
Figure 4: An illustration of and ().

Next, we consider the proof of the statement (b). Since holds for a rheonomous affine equilibrium point , we can show by setting . Hence, holds and this yields . Since the dimension of is greater than or equal to from Lemma 3.2, the -dimensional time-invariant integral manifold for the -rheonomous affine constraints (2.2): can be represented as (3.13). Moreover, holds in this case, and hence we obtain (3.14) and (3.15) from (3.11) and (3.12), respectively.

The illustrations for Theorem 3.3(a) and (b) are shown in Figures 5 and 6 for the case of , respectively. As shown in Figure 5(a), is divided into a 1-dimensional time-varying foliation structure, and a trajectory starting from an initial point at the time is constrained in the foliation . However, since the foliation slides as time goes by, the trajectory can move as a 2-dimensional behavior in . In addition, just like the illustration of the tangent space shown in Figure 5(b), lies on , however, does not belong to . This fact also causes the time-varying foliation structure.

fig5
Figure 5: An illustration for Theorem 3.3(a) ().
fig6
Figure 6: An illustration for Theorem 3.3(b) ().

On the other hand, is divided into a 1-dimensional time-invariant foliation structure as shown in Figure 6(a). Hence, a trajectory that starts from an initial point at the time is constrained in the foliation and cannot move to another foliation. Moreover, as the illustration of the tangent space shown in Figure 6(b), not only but also lies on , and hence the time-invariant foliation structure occurs.

From the statement (a) in Theorem 3.3, we can confirm the existence of time-varying foliation structures in configuration manifolds for the completely integrable -rheonomous affine constraints as is the case in the scleronomous affine constraints [23, 24]. Now, let us consider a simple example in order for graphic understanding of Theorem 3.3 as follows.

Example 3.4. We deal with the same example which is shown in the first paper [38, Example 1]. Consider a configuration manifold: with , and -rheonomous affine constraints on : with . It turns out that Assumption 2.1 holds for (3.20). One of the geometric representation for (3.20) can be obtained as follows: Calculating an iterated rheonomous brackets for and above, we obtain Hence, it turns out that all the iterated rheonomous bracket of , are 0. Therefore, we have and it turns out that holds. From Theorem 3.1, we can see that the -rheonomous affine constraints (3.20) are completely integrable. By using the integrating algorithm introduced in Section 4, we can calculate two independent first integrals of (3.20) as where is an initial point at the initial time (for the derivation of it, see Example 4.3 in Section 4).
From Theorem 3.3, the set of the rheonomous affine regular points is given by , and the set of the rheonomous affine regular points is represented by . That is to say, the - plane is coincident with and the total space of except the - plane is . The illustrations of the foliation structures for these cases are depicted in Figure 7. As shown in Figure 7(a), a trajectory starting from a rheonomous affine regular point is included in the 1-dimensional time-varying integral manifold at the time , and moves in the 2-dimensional space determined by . Therefore, has a 1-dimensional time-varying foliation structure. Meanwhile, as shown in Figure 7(b), a trajectory starting from a rheonomous affine equilibrium point is included in the 1-dimensional time-invariant integral manifold and moves only in . Consequently, has a 1-dimensional time-invariant foliation structure.

fig7
Figure 7: Foliation structures of Example 3.4.
3.2. Partially Integrable Case

Next, we investigate foliation structures of a configuration manifold for the partially integrable -rheonomous affine constraints case in this subsection. Now, necessary and sufficient conditions on partial integrability for the -rheonomous affine constraints (2.2), which are also originally derived in the first paper [38], are shown as the follows.

Theorem 3.5 (see [38]). For the -rheonomous affine constraints (2.2) defined on an -dimensional manifold and a time interval , the following statements (a)–(c) are equivalent to each other. If they hold, the -rheonomous affine constraints (2.2) are said to be th-order partially integrable or th-order partially holonomic, where .(a)There exist independent first integrals of the -rheonomous affine constraints (2.2): such that hold(b)There exists a -dimensional time-varying integral manifold of the -rheonomous affine constraints (2.2) determined by an initial point such that holds.(c)For a smallest and involutive time-varying distribution , holds.

From the statement (b) in Theorem 3.5, it turns out that there exists a -dimensional time-varying integral manifold in the -dimensional configuration manifold . Similar to the results in the previous section, we can expect that this time-varying integral manifold creates a foliation structure in . So, we investigate the foliation structure with much circumstance. We now derive the next lemma that helps to prove the main theorem.

Lemma 3.6. Assume that the -rheonomous affine constraints (2.2) are th-order completely integrable. One calls a point such that   a th order rheonomous affine regular point, and denotes a set of th-order rheonomous affine regular points by Where is a time interval. On the other hand, one call a point such that   a th-order rheonomous affine equilibrium point, and denotes a set of th-order rheonomous affine equilibrium points by Then, the following statements (a) and (b) hold.(a)For and , hold.(b)For a neighborhood of a rheonomous affine equilibrium point , holds.

Proof. Similar to the proof of Lemma 3.2, since each point in is classified into either a th-order rheonomous affine regular point or a th-order rheonomous affine equilibrium point from the definition of these points, the statement (a) obviously holds. Now, consider vector fields which are independent of and satisfy Since a th-order rheonomous affine equilibrium point satisfies , can be represented as with functions . We here consider vector fields which are independent of each other and are orthogonal to . Calculating inner products of and , we have So, since (3.35) can be considered as equations for the point , we can see that the dimension of a neighborhood of : is greater than or equal to from the implicit function theorem [39, 40]. Hence, we have completed the proof.

Hence, we now derive the main results on a foliation structure of the configuration manifold for the partially integrable -rheonomous affine constraints as follows.

Theorem 3.7. Assume that the -rheonomous affine constraints (2.2) are th order partially integrable. Then, the following statements (a) and (b) hold for a time interval .(a)For a neighborhood of a th-order rheonomous regular point : , there exist a local coordinate and a -dimensional time-varying foliation structure in which is constructed by a -dimensional time-varying integral manifold determined by the time variable and the point : In addition, hold.(b)For a neighborhood of a th-order rheonomous equilibrium point : , there exist a local coordinate and a -dimensional time-invariant foliation structure in which is constructed by a -dimensional time-invariant integral manifold determined by the point : In addition, hold.

Proof. This theorem can be proven by similar procedure to the proof of Theorem 3.1 and the results in Lemma 3.6. The details are omitted.

Figures 8 and 9 show the illustrations for Theorem 3.7(a) and (b) for the case of , , , respectively. As shown in Figure 8(a), is divided into a 2-dimensional time-varying foliation structure, and a trajectory that starts from an initial point at the time is included in the foliation . However, since the foliation slides as time goes by, the trajectory can move as a 3-dimensional behavior in . In addition, like the illustration of the tangent space shown in Figure 8(b), lies on , however, does not belong to . This result also causes the time-varying foliation structure.

fig8
Figure 8: An illustration for Theorem 3.7(a) ().
fig9
Figure 9: An illustration for Theorem 3.7(b) ().

In contrast, is divided into a 2-dimensional time-invariant foliation structure as shown in Figure 9(a), Therefore, a trajectory starting from an initial point at the time is constrained in the foliation and cannot move to another foliation. Moreover, as the illustration of the tangent space shown in Figure 9(b), as well as lies on , and hence we can find the time-invariant foliation structure in this case.

Similar to the statement (a) in Theorem 3.3, from the statement (a) in Theorem 3.7, it can be also confirmed that there exist time-varying foliation structures in configuration manifolds for the partially integrable -rheonomous affine constraints, which are in common with the case of the scleronomous affine constraints [23, 24]. We here consider a simple example in order to understand Theorem 3.7 as follows.

Example 3.8. We here consider the same example which is shown in the first paper [38, Example 2]. Let us consider a 3-dimensional configuration manifold: with and -rheonomous affine constraints on : with . We can see that Assumption 2.1 holds for (3.41). One of the geometric representation for (3.41) can be obtained as follows: By calculating iterated rheonomous brackets for and above, we have Therefore, we have and we can see that holds. From Theorem 3.3, we can find that the -rheonomous affine constraints (3.41) are 2nd-order partial nonintegrable. By using the integrating algorithm introduced in Section 4, we can obtain two independent first integrals of (3.41) as where is an initial point at the initial time (for the derivation of it, see Example 4.6 in Section 4).

From Theorem 3.7, the set of the 2nd-order rheonomous affine regular points is represented by , and the set of the 2nd-order rheonomous affine regular points is given by . Hence, the - plane corresponds to and the total space of except the - plane is . Figure 10 depicts the illustrations of the foliation structures for these cases. As shown in Figure 10(a), a trajectory starting from a 2nd-order rheonomous affine regular point is included in the 2-dimensional time-varying integral manifold at the time , and it moves in the 3-dimensional space determined by . Therefore, has a 2-dimensional time-varying foliation structure. On the other hand, as shown in Figure 10(b), a trajectory that starts from a 2nd rheonomous affine equilibrium point is included in the 2-dimensional time-invariant integral manifold and moves only in . Consequently, has a 2-dimensional time-invariant foliation structure.

fig10
Figure 10: Foliation structures of Example 3.8.
3.3. Completely Nonintegrable Case

In this subsection, we finally deal with the case where the -rheonomous affine constraints are completely nonintegrable. Now, we present necessary and sufficient conditions on complete nonintegrability for the -rheonomous affine constraints (2.2), which are originally obtained in the first paper [38] as follow.

Theorem 3.9 (see [38]). For the -rheonomous affine constraints (2.2) defined on an -dimensional manifold and a time interval , the following statements (a), (b) are equivalent to each other. If they hold, the -rheonomous affine constraints (2.2) are said to be completely nonintegrable or completely nonholonomic.(a)There do not exist independent first integrals of the -rheonomous affine constraints (2.2).(b)For a smallest and involutive time-varying distribution , holds.

From the statement (a) in Theorem 3.9, it can be easily expected that there exist no foliation structure in . We summarize this fact as the following theorem.

Theorem 3.10. Assume that the -rheonomous affine constraints (2.2) are completely nonintegrable. Then, for a time interval , holds and there is no foliation structure in . In addition, hold.

Proof. We can prove this theorem by considering the case of in the results in Theorem 3.7. The details are omitted.

For Theorem 3.10, the illustration is depicted for the case of in Figure 11. As shown in Figure 11(a), a trajectory starting from an initial point can move in without constraints by integral manifolds. Therefore, is not divided into a foliation structure. In addition, as the illustration of the tangent space shown in Figure 11(b), neither nor do not belong to and three vector fields span a 3-dimensional space. This result also causes no foliation structure in .

fig11
Figure 11: An illustration for Theorem 3.10 ().

Now, let us deal with a simple example in order to verify the results in Theorem 3.10 as follows.

Example 3.11. Now, we treat the same example which is shown in the first paper [38, Example 3]. Consider a 3-dimensional configuration manifold: with , and -rheonomous affine constraints defined on : with . We can see that Assumption 2.1 holds for (3.51). One of the geometric representation for (3.51) can be obtained: Calculating iterated rheonomous brackets for and above, we have Therefore, we have and it can be confirmed that holds. From Theorem 3.9, we can see that the -rheonomous affine constraints (3.51) are completely nonintegrable. Therefore, from Theorem 3.10, it can be confirmed that there exists no foliation structure in .

4. Integrating Algorithm for Integrable -Rheonomous Affine Constraints

4.1. Completely Integrable Case

As seen in the first paper [38] and Section 3 in this second paper, if the -rheonomous affine constraints (2.2) are completely integrable or partially integrable, there exist some independent first integrals of them, and they form time-varying and time-invariant foliation structures in a configuration manifold . If we consider reduction of the dimension of a given configuration manifold with the -rheonomous affine constraints, we need the explicit forms of independent first integrals. For scleronomous linear constraints, that is, , a method on calculation of independent first integrals of them has been well known [17, 18], and for scleronomous affine constraints, that is, , we have developed algorithms to calculate independent first integrals of them in [25]. However, a method to calculate independent first integrals of given -rheonomous affine constraints has not been proposed. Thus, in this section, we will derive two types of integrating algorithms which can explicitly derive independent first integrals of the -rheonomous affine constraints (2.2).

In this subsection, we first consider the case where the -rheonomous affine constraints (2.2) are completely integrable and derive an integrating algorithm for them. Theorem 3.1 in Section 3 guarantees the existence of independent first integrals of the -rheonomous affine constraints (2.2). Now, we derive an algorithm to calculate the independent first integrals. First of all, we can find vector fields such that holds for the vector fields of the geometric representation for the -rheonomous affine constraints (2.2): . Then, let us denote flows (1-parameter local transformation groups) of and by and with parameters and , respectively. We set an initial point at the initial time as . We also consider vector fields defined on the expanded configuration manifold (3.18), and then their flows on are obtained as Calculating the composite mapping of flows (4.15) yields where is the composite mapping of flows . From (4.16), it turns out that the projection of onto is equivalent to . Hence, by applying the idea of the integrating algorithm for scleronomous linear constraints defined on [17, 18] to and considering projection of it onto , we can derive the following algorithm to calculate independent first integrals of completely integrable -rheonomous affine constraints.

Algorithm 4.1. For the case where the -rheonomous affine constraints (2.2) are completely integrable, we can obtain independent first integrals of them by the following procedure.

Step 1. Set vector fields of geometric representation for the -rheonomous affine constraints (2.2).

Step 2. For , derive linearly independent vector fields that satisfy (4.14).

Step 3. Calculate flows of : .

Step 4. Combine flows derived in Step 4 as (4.17).

Step 5. Set and calculate the inverse function , where . Then, the last components of are independent first integrals of (2.2).

It must be noted that Algorithm 4.1 is similar to the one for the scleronomous affine constraints case [25] despite the explicit existence of the time variable in the -rheonomous affine constraints (2.2). Moreover, the existence of the inverse mapping which appears at Step 5 in Algorithm 4.1 is guaranteed by the following proposition.

Proposition 4.2. Assume that the -rheonomous affine constraints (2.2) are completely integrable. Then, there exists a time interval and is a diffeomorphism at any time . That is to say, there exists its inverse mapping .

Proof. First, we set . Calculating the partial differential of (4.16) with the chain rule of differential calculation, we have Substituting into (4.5), we obtain that is to say, Since are linearly independent of each other, holds. Hence it can be confirmed that is a diffeomorphism by the implicit function theorem. Since the projection of onto is equivalent to , is also a diffeomorphism. Consequently, the proof of the proposition is completed.

Now, we deal with a simple example in order to check the effectiveness of Algorithm 4.1 and Proposition 2.7 as follows.

Example 4.3. We here consider the -rheonomous affine constraints (3.50) on the 3-dimensional configuration manifold (3.20), which are the same ones in Example 3.4. As seen in Example 3.4, we can confirm that (3.50) are completely integrable with , and hence there exist two independent first integrals of (3.50). So, let us calculate them by using Algorithm 4.1. One of the geometric representation for (3.50) is given by (3.21) and we here set . We can easily find two vector fields satisfying (4.14) as and then flows of , ,, can be obtained as where is an initial point of the flows at the initial time. Combining the flows (4.10) as (4.17), we have and from the equation , we can calculate the inverse mapping of as follows: Consequently, we have two independent first integrals of (3.50): as the last two components of (4.12). It can be also confirmed that these two first integrals (4.13) satisfy (3.1) and hence (4.13) are independent of each other.

4.2. Partially Integrable Case

As seen in the first paper [38] and Section 3 in this second paper, if the -rheonomous affine constraints (2.2) are completely integrable or partially integrable, there exist some independent first integrals of them, and they form time-varying and time-invariant foliation structures in a configuration manifold . If we consider reduction of the dimension of a given configuration manifold with the -rheonomous affine constraints, we need the explicit forms of independent first integrals. For scleronomous linear constraints, that is, , a method on calculation of independent first integrals of them has been well known [17, 18], and for scleronomous affine constraints, that is, , we have developed algorithms to calculate independent first integrals of them in [25]. However, a method to calculate independent first integrals of given -rheonomous affine constraints has not been proposed. Thus, in this section, we will derive two types of integrating algorithms which can explicitly derive independent first integrals of the -rheonomous affine constraints (2.2).

In this subsection, we first consider the case where the -rheonomous affine constraints (2.2) are completely integrable and derive an integrating algorithm for them. Theorem 3.1 in Section 3 guarantees the existence of independent first integrals of the -rheonomous affine constraints (2.2). Now, we derive an algorithm to calculate the independent first integrals. First of all, we can find vector fields such that holds for the vector fields of the geometric representation for the -rheonomous affine constraints (2.2): . Then, let us denote flows (1-parameter local transformation groups) of and by and with parameters and , respectively. We set an initial point at the initial time as . We also consider vector fields defined on the expanded configuration manifold (3.18), and then their flows on are obtained as Calculating the composite mapping of flows (4.15) yields where is the composite mapping of flows . From (4.16), it turns out that the projection of onto is equivalent to . Hence, by applying the idea of the integrating algorithm for scleronomous linear constraints defined on [17, 18] to and considering projection of it onto , we can derive the following algorithm to calculate independent first integrals of completely integrable -rheonomous affine constraints.

Algorithm 4.4. For the case where the -rheonomous affine constraints (2.2) are th-order partially integrable, we can obtain independent first integrals of them by the following procedure.

Step 1. Set vector fields of geometric representation for the -rheonomous affine constraints (2.2).

Step 2. By calculating iterated Lie brackets of , derive linearly independent vector fields that satisfy

Step 3. For , derive linearly independent vector fields that satisfy

Step 4. Calculate flows of : .

Step 5. Combine flows derived in Step 4 as (4.17).

Step 6. Set and calculate the inverse function . Then, the last components of are first integrals of (2.2).

It is noteworthy that the differences between Algorithms 4.1 and 4.4 appear in Steps 2 and 6 in Algorithm 4.4, that is, we have to set vector fields that satisfies (4.18) and the number of independent first integrals obtained by Algorithm 4.4 is smaller than the ones obtained by Algorithm 4.1. Similar to Proposition 2.3, the existence of the inverse mapping at Step 5 in Algorithm 4.4 is guaranteed by the following proposition.

Proposition 4.5. Assume that the -rheonomous affine constraints (2.2) are th-order partially integrable. Then, there exists a time interval and is a diffeomorphism at any time . That is, there exists its inverse mapping .

Proof. This proposition can be proven by similar procedure to the proof of Proposition 2.7. The details are omitted.

Now, let us consider another simple example to check the availability of Algorithm 4.4 and Proposition 4.2 as follows.

Example 4.6. We now deal with the -rheonomous affine constraints (3.41) on the 3-dimensional configuration manifold (3.40), which are the same ones in Example 3.8. As seen in Example 3.8, we can find that (3.41) are 2nd-order partially integrable with , there exists one independent first integrals, and hence we now calculate them by using Algorithm 4.4. One of the geometric representation for (3.41) is given by (3.42) and we set . Calculating iterated rheonomous brackets for (3.42), we have , where Next, we set and calculate satisfying (4.19) as and the flows of , , , can be obtained as where is an initial point of the flows at the initial time. Combining (4.22) as (4.17), we have and from the equation , we can calculate its inverse mapping: Consequently, we can obtain one independent first integral of (3.41): as the last component of (4.24).

5. Physical Example: Pendulum with Time-Varying Elastic String

In this section, we consider a physical example in order to verify the results obtained in the previous sections. We treat a pendulum with an elastic string, which is shown in Figure 12 and has already been dealt with in the first paper [38]. This system is composed of a weight and an elastic string. One end of the string is fixed to the ceiling and can swing around the fixed point. The weight is installed at the other end of the string. As depicted in Figure 12, we set the -axis and the -axis to the horizontal and vertical directions, respectively. We also set the origin of the coordinate system so that it corresponds to the fixed point of the string. Therefore, we consider the 2-dimensional configuration manifold as follows: where . The position of the weight in is represented by . In addition, the angle of the string in the clockwise direction measured from the -axis is denoted by . Now, we assume that the length of the string is changed as time goes by, that is, a time-varying elastic string, and we denote it by .

345942.fig.0012
Figure 12: A single pendulum with a time-varying elastic string.

This system is undoubtedly subject to a constraint on the configuration variables as We can easily see that since (5.2) does not contain any derivatives of the configuration variables, (5.2) is a rheonomous holonomic constraint. So, we can reduce the number of the configuration variables by replacing by . However, we consider this system as a physical example which is subject to an integrable -rheonomous affine constraint. Calculating time derivative of (5.2), we have Now, for the sake of simplicity, we set . Then, (5.3) can be represented as an -rheonomous affine constraint defined on : where . Let us start with the above -rheonomous affine constraint (5.4). One of geometric representation for the -rheonomous affine constraints (5.4) can be derived as