Research Article | Open Access

Volume 2012 |Article ID 543098 | https://doi.org/10.1155/2012/543098

Tatsuya Kai, "Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part I: Fundamental Properties and Integrability/Nonintegrability Conditions", Mathematical Problems in Engineering, vol. 2012, Article ID 543098, 32 pages, 2012. https://doi.org/10.1155/2012/543098

Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part I: Fundamental Properties and Integrability/Nonintegrability Conditions

Accepted04 Jul 2012
Published13 Aug 2012

Abstract

We analyze a class of rheonomous affine constraints defined on configuration manifolds from the viewpoint of integrability/nonintegrability. First, we give the definition of A-rheonomous affine constraints and introduce, geometric representation their. Some fundamental properties of the A-rheonomous affine constrains are also derived. We next define the rheonomous bracket and derive some necessary and sufficient conditions on the respective three cases: complete integrability, partial integrability, and complete nonintegrability for the A-rheonomous affine constrains. Then, we apply the integrability/nonintegrability conditions to some physical examples in order to confirm the effectiveness of our new results.

1. Introduction

In the research fields such as control theory and robotics, nonholonomic systems have been actively studied by a lot of researchers so far [15]. Nonholonomic systems are, in the simplest terms, defined as ones that are subject to nonintegrable constraints and whose behaviors must satisfy the constraints. That is to say, the word “nonholonomic” is equivalently used as “nonintegrable.” We can easily find a lot of examples of nonholonomic systems: 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], and so on. In past work on nonholonomic systems, linear constraints have been mainly treated. The linear constraints are represented as and contain various examples such as mobile cars, a ball or a coin on a plain, and hopping robots [14, 15]. Integrability and nonintegrability of the linear constraints can be determined by using the well-known Frobenius’ theorem [17, 18], and they are strongly related to accessibility of nonlinear control systems that are derived from the linear constraints [4, 5, 7, 1721]. Namely, researches on systems subject to the linear constraints have been highly developed by various approaches so far. Especially, researches on nonholonomic control systems can be mainly classified into the two research fields: kinematic systems and dynamic systems. Kinematic systems are directly derived from nonholonomic constraints, and in particular the linear constraints can be transformed into symmetrically affine control systems [7, 22, 23]. On the contrary, dynamic systems are derived from Euler-Lagrange equations with the constrained forces based on d'Alembert's principle [21, 24, 25]. There are two common characteristics for both kinematic and dynamic systems: (i) their linear approximated systems are uncontrollable and (ii) they are locally controllable, but not locally asymptotically stabilizable by any nonlinear smooth state feedback laws from Brockett's theorem [4, 26]. Therefore, a lot of control methods such as time-variant feedback, discontinuous feedback, and switching control laws have been proposed to avoid Brockett's condition.

On the other hand, in nonholonomic systems, there exists another class of constraints called affine constraints. The affine constraints are represented as , , and contain some curious examples such as space robots with initial angular momenta (Figure 1(a)), a boat or a ship on a running river (Figure 1(b)), and a ball or a coin on a rotating table (Figures 1(c) and 1(d)). Obviously, the class of the affine constraints is larger than that of the linear constraints and covers a wide range of nonholonomic systems; however, theoretical analysis on the affine constraints had been hardly done. So, we focused on and researched the affine constraints from the viewpoints of both mathematics and control theory and derived various results: integrability/nonintegrability conditions of the affine constraints and foliation structures of configuration manifolds [27], integrating algorithms of integrable affine constraints [28], nonlinear control analysis and control of nonholonomic kinematic systems with affine constraints [2932], modeling and theoretical analysis of nonholonomic dynamic systems with affine constraints [31, 33], modeling and passivity analysis of nonholonomic Hamiltonian systems with affine constraints [34], generalized canonical transformation and passivity-based control of nonholonomic Hamiltonian systems with affine constraints [35], and near-optimal control of a 3D space robot with an initial angular momentum [36]. In a sequence of researches on the affine constraints, we obtained some important results: (a) complete nonintegrability (nonholonomicity) of the affine constraints is equivalent to strong accessibility of systems, (b) linear approximated systems of original systems with affine constraints are controllable under some conditions, and (c) some systems with affine constraints satisfy Brockett's condition and hence there is a possibility of local asymptotic stabilizability by nonlinear smooth state feedback laws. We can say that these characteristics are beyond the ones for the linear constraints and indicate a new control problem for nonholonomic control systems.

All the constraints that are dealt with in the researches above do not contain the time variable, that is, scleronomous constraints [1]. However, there exist nonholonomic mechanical systems whose constraints contain the time variable. For example, 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 physical systems that are subject to affine constraints, which contain the time variable. These constraints are called rheonomous [1], especially, the class of -rheonomous affine constraints that are represented as , ,  , , that is, affine constraints whose term contains the time variable is important from an application standpoint. Moreover, the class of the -rheonomous affine constraints is larger than that of the scleronomous affine constraints, and so we can say that the range of possible application extends more and more as shown in Figure 2. For analysis and control synthesis of mechanical systems subject to the rheonomous affine constraints, it is essential to consider them; however, the existing results on scleronomous affine constraints [27, 29] cannot be utilized. Hence, we need a new fundamental theory on the rheonomous affine constraints such as integrability/nonintegrability and foliation structures of configuration manifolds in order to apply control theory to such systems, and it is expected that the existence of the time variable in the constraints makes it more difficult to analyze them. In [3741], rheonomous constraints have been dealt with in terms of Lagrangian mechanics, jet bundles, and almost-Poisson structures, but fundamental properties of rheonomous constraints such as integrability/nonintegrability and foliation structures of configuration manifolds have not been considered in these works.

This is the first paper of a series of papers on theoretical analysis of the -rheonomous affine constraints defined on configuration manifolds. In this first paper, we aim at derivation of fundamental properties and integrability/nonintegrability conditions for the -rheonomous affine constraints. specifically, integrability/nonintegrability analysis of the -rheonomous affine constraints is one of the most basic and important concepts. On the other hand, the second paper [42] focuses on analysis of foliation structures of configuration manifolds with the -rheonomous affine constraints and development of integrating algorithms for integrable -rheonomous affine constraints.

This paper is organized as follows. In Section 2, we first give a definition of the -rheonomous affine constraints and explain their geometric representation. we also derive some fundamental properties of the -rheonomous affine constraints. Next, Section 3 introduces the rheonomous bracket that is a new operator and plays an important role, and presents necessary and sufficient conditions for the -rheonomous affine constraints in the three cases: complete integrability, partial integrability, and complete nonintegrability. Then, we apply our new results to some physical examples in order to confirm their effectiveness. Throughout this paper, manifolds, submanifolds, functions, vector fields, distributions, and differential forms are all assumed to be smooth.

2. Rheonomous Affine Constraints

2.1. Definition

In this subsection, we give a definition of rheonomous affine constraints. We denote the time variable by and a time interval by . Let be an -dimensional configuration manifold and to a local coordinate of . Associated with , we refer as a tangent vector field.

A set of differential equations: is called rheonomous affine constraints because the coefficients depend on the time variable . We now 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 .

In this paper, we consider a subclass of the rheonomous affine constraints (2.2) in the form that is to say, only the rheonomous affine term depends on the time variable . We call (2.3) -rheonomous affine constraints. It must be noted that this class of the -rheonomous affine constraints contains some important examples of mechanical systems as mentioned in Section 1. Now, we assume a sufficient condition on independency of the -rheonomous affine constraints (2.3) as follows.

Assumption 2.1. The coefficient matrix of the -rheonomous affine constraints (2.3) has a row full-rank at any point , that is, holds.

Then, we here give a definition on classification of points in the configuration manifold in terms of the rheonomous affine term in the -rheonomous affine constraints (2.3) as follows.

Definition 2.2. For the rheonomous affine term of the -rheonomous affine constraints (2.3) 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.

From Definition 2.2, we can see that each point in is classified into either a rheonomous affine regular point or a rheonomous affine equilibrium point.

2.2. Geometric Representation and Some Properties

This subsection introduces a geometric representation method and provides some fundamental properties of the -rheonomous affine constraints.

From (2.4) in Assumption 2.1, the row vectors of in the -rheonomous affine constraints (2.3) are independent of each other at any point . Hence, we here consider vectors that are independent of each other and annihilators of the row vectors of and denote them by as vector fields on . In addition, 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.

In order to represent the -rheonomous affine constraints geometrically, we now introduce an important vector field on . A curve is said to be satisfying the -rheonomous affine constraints (2.3) if for a vector field on : and the generalized velocity of : , holds as shown in Figure 3. is called a rheonomous affine vector field. This definition is an extension of the one for the scleronomous affine constraints that do not contain the time variable [27].

Now, we show essential characteristics of the rheonomous affine vector field in the following proposition.

Proposition 2.3. For the -rheonomous affine constraints (2.3), the component of the rheonomous affine vector field , and a time interval , holds.

Proof. Assume that a velocity vector at a point satisfies the rheonomous affine constraints (2.3). Since holds, we obtain where are functions on . Now, multiply the left-hand side of (2.9) by . Since the row vectors of are annihilators of , we then have Furthermore, using (2.3), we can rewrite (2.10) as and hence we obtain (2.8).

Consequently, under the preliminaries shown above, the -rheonomous affine constraints (2.3) can be geometrically represented as the next definition.

Definition 2.4. The -rheonomous affine constraints (2.3) are geometrically represented by a pair , where is an -dimensional distribution defined by (2.5) and is called a rheonomous affine vector field and satisfies (2.8).

In order to derive some conditions on integrability/nonintegrability for -rheonomous affine constraints (2.3), their geometric representation 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. For the geometric representation of the -rheonomous affine constraints (2.3); 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.

Proof. If is a rheonomous affine equilibrium point, that is, holds, we have and hence , is derived from (2.3). Conversely, if holds at a point , that is, , then we have from (2.3), and it turns out that the point is a rheonomous affine equilibrium point. Hence, a point is a rheonomous equilibrium point if and only if , holds.
Next, all the points in are classified into either rheonomous affine regular points or rheonomous affine equilibrium points from Definition 2.2. Therefore, a point is a rheonomous regular point if and only if holds. Consequently, we have completed the proof of this proposition.

3. Integrability and Nonintegrability Conditions

3.1. Rheonomous Bracket

This section presents some necessary and sufficient conditions for the -rheonomous affine constraints (2.3) in the three cases of complete integrability, partial nonintegrability, and complete nonintegrability. First of all, we give a definition of a new operator called rheonomous bracket in this subsection. The normal Lie bracket for two vector fields is defined as Now, in order to derive some conditions for the -rheonomous affine constraints, we define a new bracket, which can be interpreted as an extension of the normal Lie bracket above, as follows.

Definition 3.1. 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.

In Definition 3.1, it is the main characteristic that the rheonomous affine vector field is perceived as special, and this yields an additional term of a time differential of a vector field as shown in property (b). The rheonomous bracket will play important roles in not only derivation of integrability and nonintegrability conditions for the -rheonomous affine constraints in the next subsections of this first paper but also analysis of foliation structure of configuration manifolds and development of integrating algorithms for the -rheonomous affine constraints in the second paper [42]. It must be also noted that from Definition 3.1 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. In addition, it turns out from the next proposition that the rheonomous bracket has the important characteristics in common with the normal Lie bracket.

Proposition 3.2. 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:

Proof. Based on the definition of the rheonomous bracket, we can calculate Hence, we complete the proof of (a). Next, a simple calculation can show that Therefore, (b) holds. Finally, we will prove (c). Since we can calculate we obtain where we utilize Jacobi's identity for the normal Lie bracket. Consequently, the proof of (c) is completed.

The properties in Proposition 2.5 can reduce the effort to calculate iterated rheonomous brackets in checking complete nonholonomicity of the given -rheonomous affine constraints.

3.2. Complete Integrability Conditions

In this subsection and the next two subsections, we will investigate the integrability/nonintegrability of the -rheonomous affine constraints (2.3). First, in this subsection, we consider the case where all the   -rheonomous affine constraints (2.3) can be integrated, that is to say, the complete integrability case.

If the -rheonomous affine constraints (2.3) are completely integrable, we can integrate them and obtain their independent first integrals that are represented by only the time variable and the configuration variable and do not contain any differentials of the configuration variable. Since we can reduce the dimension of a system subject to constraints by using independent first integrals of them and transform the system into another system with no constraints, it is quite important to examine if the constraints are integrable. We now define a smallest and involutive time-varying distribution which contains and iterated rheonomous brackets of them, and satisfies , that is to say, is spanned by all the rheonomous brackets of with the exception of . Then, some necessary and sufficient conditions on complete integrability for the -rheonomous affine constraints (2.3) are given by the following theorem.

Theorem 3.3. For the -rheonomous affine constraints (2.3) 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 (2.3) are said to be completely integrable or completely holonomic.(a)There exist independent first integrals of the -rheonomous affine constraints (2.3): 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 (2.3) determined by an initial point such that holds.(c)For a smallest and involutive time-varying distribution , holds.

Proof. First of all, the setting for the proof is given. We denote the product manifold of the space of the time variable and the configuration manifold by with the coordinate , and is called an expanded configuration manifold with the dimension . Hence, the -rheonomous affine constraints (2.3) defined on can be represented by an expanded Pfaffian equation on : with expanded differential forms: Since the rheonomous affine vector field satisfies and the vector fields satisfy , annihilators of the expanded differential forms (3.18) can be derived as We consider an -dimensional expanded distribution on that is spanned by the vector fields and denote it by We will prove this theorem in line with the diagram shown in Figure 4. Since it is possible to show that the statements (i), (ii), and (iii) in Figure 4 are equivalent to each other on by Frobenius' theorem [17, 19], all we have to do is prove the three equivalences between (a) and (i), (b) and (ii), and (c) and (iii).
We begin with the proof of the equivalence between (a) and (i). First, we prove (a) (i). Assume that (a) holds, that is, there exist independent first integrals of the -rheonomous affine constraints (2.3) on and they satisfy (3.12)–(3.14). Now, we define and show that these are independent first integrals of the expanded Pfaffian equation (3.17) on , that is, hold, where is the expanded exterior derivative operator defined on . From the form of and (3.12), we have From the form of and (3.13), we also obtain Therefore, it turns out that (3.21) holds. Furthermore, calculating the left-hand side of (3.22), we have and hence we can see that (3.22) holds under (3.14). So, the proof of (a) (i) is completed. Conversely, we then prove (a) (i). It is assumed that there exist independent first integrals of the expanded Pfaffian equation (3.17) on , , and they satisfy (3.21) and (3.22). We define and show that they are independent first integrals of the -rheonomous affine constraints (2.3) on , that is to say, (3.12) and (3.13) hold. By using (3.21), we obtain and hence (3.12) holds. We now assume that , that is, are not independent of each other. Then, without loss of generality, we can write that is to say, is represented as a linear combination of with coefficients . We then have Therefore, we can find that holds, and hence (3.29) is inconsistent with assumption (3.22). So, it can be confirmed that holds and are independent first integrals. Consequently, we have proven (a) (i), and hence the proof of (a) (i) is finished.
Next, we prove the equivalence between (b) and (ii). We first prove (b) (ii). See also the illustration shown in Figure 5 for understanding the proof. Assume that for an -dimensional submanifold in with the initial point : , holds. Now, we denote a diffeomorphism that is a projection from to at the time by and its derivation by . Consider a decomposition of as , where . In addition, we set , C and the projection of on to by is defined as . Then, we can find that holds, and hence from (3.19) and we derive Therefore, from (3.30), (3.15) holds and we conclude (b) (ii). In contrast, we can prove (b) (ii) by using the inverse mapping of : in the same manner. Especially, from on , we can construct an -dimensional submanifold on as So, the proof of (b) (ii) is completed, and hence we conclude that (b) (ii) holds.
Finally, we give a proof on the equivalence between (c) and (iii). We here define an involutive distribution defined on , which contains and iterated Lie brackets that consist of . Therefore, a necessary and sufficient condition of the complete integrability for the expanded Pfaffian equation is given by (cf. Frobenius' theorem [17, 19]). Calculating the iterated Lie brackets that consist of , we have We can see that is independent of and the iterated Lie brackets (3.36). Then, the necessary and sufficient condition (3.35) is changed into the condition such that and the iterated Lie brackets which consist of span an -dimensional space. From (3.19) and (3.36), we can consider only on instead of on and iterated rheonomous brackets that consist of on instead of Lie brackets that consist of on . Therefore, a necessary and sufficient condition of complete integrability for the rheonomous affine constraints (2.2) is that and the iterated rheonomous brackets which consist of span an -dimensional space, that is, (3.16) holds. Consequently, we complete the proof of (c) (iii).
Since we have proved the equivalence between (a) and (i), (b) and (ii), and (c) and (iii), we have completed the proof of this theorem of the complete integrability of the -rheonomous affine constraints (2.3).

It must be noted that though the -rheonomous affine constraints (2.3) contain the time variable, conditions (a)–(c) in Theorem 3.3 are similar to the ones for the scleronomous affine constraints case [27, 29], and the rheonomous bracket plays an essential role. In order to the check complete integrability of the given -rheonomous affine constraints, the condition (c) in Theorem 3.3 is very useful. By calculating some iterated rheonomous brackets of vector fields of their geometric representation and investigating the dimension of , we can easily determine their complete integrability. Now, we consider a simple example in order to verify Theorem 3.3 as follows.

Example 3.4. We consider a 3-dimensional configuration manifold with and -rheonomous affine constraints on with . In addition, we consider a time interval . We can see that Assumption 2.1 holds for (3.38). One of the geometric representations of (3.38) can be obtained as follows: Calculating a iterated rheonomous brackets for and above, we obtain Hence, it turns out that all the iterated rheonomous brackets of and are 0. Therefore, we have and it is confirmed that holds. From Theorem 3.3, we can see that the -rheonomous affine constraints (3.38) are completely integrable. In fact, there exist two independent first integrals of (3.38): where is an initial point at the initial time. We can easily confirm that (3.43) satisfies (3.12)–(3.14) of condition (a) in Theorem 3.3, and hence (3.43) are independent first integrals of (3.38). In the second paper [42], we will derive an integrating algorithm for completely integrable -rheonomous affine constraints and show how to calculate (3.43).

3.3. Partial Integrability Conditions

In this subsection, we next consider the case where some of the -rheonomous affine constraints in (2.3) are integrable and the others are nonintegrable, that is to say, integrable constraints and nonintegrable ones are mixed. We call such a case partial integrability. Now, let us derive some necessary and sufficient conditions on the partial integrability of the -rheonomous affine constraints (2.3) in the following theorem.

Theorem 3.5. For the -rheonomous affine constraints (2.3) 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 (2.3) 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 (2.3): such that hold.(b)There exists a -dimensional time-varying integral manifold of the -rheonomous affine constraints (2.3) determined by an initial point such that holds.(c)For a smallest and involutive time-varying distribution , (d)holds.

Proof. We prove this theorem along the diagram shown in Figure 6. Since similar to the proof of Theorem 3.3, we can show that the statements (i)–(iii) in Figure 6 are equivalent to each other by using Frobenius' theorem [17, 19], we will prove the three equivalences between (a) and (i), (b) and (ii), and (c) and (iii) to complete the proof.
First, we show the equivalence between (a) and (i). Define a new smallest and involutive distribution on , , that contains all the iterated Lie brackets of and iterated Lie brackets their and satisfies for . That is to say, is a subbundle in that is spanned by ( is not included) and iterated Lie brackets of . Then, we can see that holds. We now prove (a) (i). We assume that there exist independent first integrals of the -rheonomous affine constraints (2.3) on : and they satisfy (3.44)–(3.46). We only have to show that functions on : are independent first integrals of the expanded Pfaffian equation (3.17) on , that is, hold. From (3.44), it turns out that holds. In addition, we can see that holds for from (3.45). Hence, by using (3.49), we obtain (3.50). By contrast, for (a) (i), since holds, we have (3.51) under assumption (3.46). Therefore, it can be confirmed that are independent first integrals of the expanded Pfaffian equation (3.18) on . Consequently, we have proven (a) (i).
We next show the equivalence between (b) and (ii) and begin with the proof of (b) (ii). Denote by a ()-dimensional submanifold on with an initial point and assume that holds. By using the mapping , which is introduced in the proof of Theorem 3.3, from (3.49) we obtain and hence from (3.53) we can show (3.47). Conversely, we can prove (b) (ii) by using the inverse mapping . So, the proof on (b) (ii) is completed.
Finally, we give the proof of the equivalence between (c) and (iii). We now prove (c) (iii). Since holds from (3.48), we set where are independent vector fields on that are independent of . Define vector fields on as . From (3.19) and (3.36), we can see that holds and hence we have . This results in (c) (iii). For (c) (iii), the proof can be done by the inverse logic of the proof of (c) (iii). Hence, we have completed (c) (iii).
As a result, from the discussion above, we conclude that the statements (a)–(c) are equivalent to each other.

Note that similar to Theorem 3.3, conditions (a)–(c) in Theorem 3.5 are also analogous to the ones for the scleronomous affine constraints case [27, 29], and the role of the rheonomous affine bracket is quite important here. Moreover, condition (c) in Theorem 3.5 is also quite useful for checking the partial integrability of the given -rheonomous affine constraints. In order to confirm Theorem 3.5, a simple example is now shown as follows.

Example 3.6. Consider a 3-dimensional configuration manifold with and -rheonomous affine constraints on with . Moreover, we consider a time interval . We can see that Assumption 2.1 holds for (3.58). One of the geometric representations of (3.58) can be obtained as follows: By calculating the iterated rheonomous brackets of and above, we have Therefore, we have and we can see that holds. From Theorem 3.5, we can find that the -rheonomous affine constraints (3.58) are 2nd order partial nonintegrable with . In fact, there exists one first integral of (3.58): where is an initial point at the initial time. We can easily confirm that (3.63) satisfies (3.12)–(3.14) of condition (a) in Theorem 3.5, and hence (3.63) are independent first integrals of (3.58). In the second paper [42], we will show how to derive (3.63) by using an integrating algorithm for partially integrable -rheonomous affine constraints.

3.4. Complete Nonintegrability Conditions

Finally, we deal with the remaining case where all the -rheonomous affine constraints (2.3) are nonintegrable. This case is called complete nonintegrability. The complete nonintegrability case is quite important in nonholonomic control theory and sub-Riemannian geometry [43, 44]. Some necessary and sufficient conditions on the complete nonintegrability of the -rheonomous affine constraints (2.3) are shown in the next theorem.

Theorem 3.7. For the -rheonomous affine constraints (2.3) defined on an -dimensional manifold and a time interval , the following statements (a) and (b) are equivalent to each other. If they hold, the -rheonomous affine constraints (2.3) are said to be completely nonintegrable or completely nonholonomic.(a)There do not exist independent first integrals of the -rheonomous affine constraints (2.3).(b)For a smallest and involutive time-varying distribution , holds.

Proof. The diagram of the proof of Theorem 3.7 is shown in Figure 7. Similar to the proofs of Theorems 3.3 and 3.5, we can show that the statements (i) and (ii) in Figure 7 are equivalent to each other by using Frobenius' theorem [17, 19], and we only have to prove the two equivalences between (a) and (i), and (b) and (ii) in order to complete the proof of this theorem. First, we can find that (a) (i) obviously holds. Then, considering the case where in the proof of Theorem 3.5, we can also prove (b) (ii), and the details are omitted. Consequently, we conclude that the statements (a) and (b) are equivalent to each other.

From the results in Theorems 3.33.7, we can conclude that the integrability or nonintegrability of the given -rheonomous affine constraints can be confirmed by calculating a smallest and involutive time-varying distribution with the rheonomous bracket and the conditions are similar to the ones for the scleronomous affine constraints case [27, 29] despite the explicit existence of the time variable in the -rheonomous affine constraints. Now, we illustrate a simple example to check the availability of Theorem 3.7 as follows.

Example 3.8. We here consider a 3-dimensional configuration manifold with and -rheonomous affine constraints on with . Now, we consider a time interval . We can see that Assumption 2.1 holds for (3.66). One of the geometric representations of (3.66) can be obtained as follows: Calculating the iterated rheonomous brackets of and above, we have Therefore, we have and it can be confirmed that holds. From Theorem 3.7, we can see that the -rheonomous affine constraints (3.66) are completely nonintegrable, and hence there exist no independent first integrals of (3.66).

4. Physical Examples

4.1. Pendulum with Time-Varying Elastic String

In this section, we consider three types of physical examples in order to confirm the results obtained in the previous sections.

First, in this subsection, we deal with a pendulum with an elastic string as shown in Figure 8. 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 shown in Figure 8, 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 -dimensional configuration manifold with . The position of the weight in is represented by , and 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 the length by .

Obviously, this system is subject to a constraint on the configuration variables as It is easily confirmed that since (4.2) does not contain any derivatives of the configuration variables, (4.2) is a rheonomous holonomic constraint. Hence, we can reduce the number of the configuration variables by replacing by . However, we here deal with this system as a physical example that is subject to an integrable -rheonomous affine constraint. Calculating time derivative of (4.2), we have Now, for the sake of simplicity, we set . Then, (4.3) can be represented as an -rheonomous affine constraint: where . We start with the above -rheonomous affine constraint the (4.4). One of geometric representations of the -rheonomous affine constraints (4.4) can be derived as For the vector fields and , we calculate a rheonomous bracket as and hence we can see that all the iterated rheonomous brackets of and are 0. Therefore, we have and so holds. From Theorem 3.3, it turns out that the -rheonomous affine constraint (4.4) is completely integrable, and this result coincides with the problem setting of this system.

4.2. Boat on Running River with Time-Varying Stream

Next, we deal with another physical example called a boat on a running river as shown in Figure 9. Set the -axis and the -axis to the transverse direction and the downstream direction of the river, respectively, and denote the center of inertia of the boat by . In addition, let be the angle of the boat. Let be a stream of the river that depends on the time variable as well as the transverse position , that is, the stream changes as time goes by. It is assumed that the boat is affected by the stream to the downstream direction of the river according to the angle of the boat , and hence the boat drifts to the -direction. Then, the generalized coordinate of this system is represented by , and we consider the -dimensional configuration manifold with , where is the length of the river.

Considering the balance of the velocities in both the heading and side directions of the boat, we have the -rheonomous affine constraints of this system as where . We can derive one of the geometric representations of the -rheonomous affine constraints (4.10) as For the vector fields , calculate some iterated rheonomous brackets: