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

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 [1–5] 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 [14–16], among others. In a lot of researches on nonholonomic control systems, linear constraints, which are represented in the form: 𝐵(𝑞)𝑞̇=0, 𝑞∈𝐑^𝑛, 𝐵(𝑞)∈𝐑^{(𝑛−𝑚)×𝑛}, 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, 17–21]. 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 𝐴(𝑞)+𝐵(𝑞)𝑞̇=0, 𝑞∈𝐑^𝑛, 𝐴(𝑞)∈𝐑^𝑛−𝑚, 𝐵(𝑞)∈𝐑^{(𝑛−𝑚)×𝑛}. 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 [23–32]. 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 [33–37] 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.

(a) Space robot with initial angular momentum

(b) Boat on running river

(c) Ball on rotating table

(d) Coin on rotating table

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 𝐴-rheonomous affine constraints defined on configuration manifolds. The 𝐴-rheonomous affine constraints are represented in the form 𝐴(𝑡,𝑞)+𝐵(𝑞)𝑞̇=0,𝑞∈𝐑^𝑛,𝐴(𝑡,𝑞)∈𝐑^𝑛−𝑚,𝐵(𝑞)∈𝐑^{(𝑛−𝑚)×𝑛}, that is to say, affine constraints whose term 𝐴 contains the time variable 𝑡. The class of the 𝐴-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 𝐴-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 𝐴-rheonomous affine constraints, and development of integrating algorithms to calculate independent first integrals for integrable 𝐴-rheonomous affine constraints.

Figure 2 

Classes of constraints and examples.

The outline of this paper is as follows. First, we give some preliminaries on the 𝐴-rheonomous affine constraints in Section 2. In Section 3, we next investigate the relationships between the 𝐴-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 𝐴-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 𝑡∈𝐑and a time interval by 𝐼⊂𝐑. Let 𝑄 be an 𝑛-dimensional configuration manifold and 𝑞=[𝑞₁⋯𝑞_𝑛]^⊤∈𝐑^𝑛 be a local coordinate of 𝑄. Associated with 𝑞, we refer 𝑞̇=[𝑞̇₁⋯𝑞̇_𝑛]^⊤∈𝑇_𝑞𝑄 as a tangent vector field.

Throughout this paper, we deal with 𝐴-rheonomous affine constraints: 𝐴_𝑖𝑡,𝑞+𝐵_𝑖1𝑞𝑞̇₁+⋯+𝐵_𝑖𝑛𝑞𝑞̇_𝑛=0,𝑖=1,…,𝑛−𝑚. (2.1) Note that the coefficients 𝐴_𝑖(𝑖=1,…,𝑛−𝑚) in (2.1) depend on the time variable 𝑡. We rewrite (2.1) as 𝐴𝑡,𝑞+𝐵𝑞𝑞̇=0, (2.2) where a rheonomous affine term 𝐴(𝑡,𝑞)∈𝐑^𝑛−𝑚 is a vector-valued function whose 𝑖th entry is 𝐴_𝑖(𝑡,𝑞), and 𝐵(𝑞) is a matrix-valued function whose 𝑖𝑗th entry is 𝐵_𝑖𝑗(𝑞). It must be noted that the class of the 𝐴-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 𝐴-rheonomous affine constraints (2.2) as follows.

Assumption 2.1. The coefficient matrix 𝐵(𝑞) of the 𝐴-rheonomous affine constraints (2.2) has a row full-rank at any point 𝑞∈𝑄, that is, rank𝐵𝑞=𝑛−𝑚,∀𝑞∈𝑄, (2.3) holds.

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

Definition 2.2 (see [38]). For the rheonomous affine term 𝐴(𝑡,𝑞) of the 𝐴-rheonomous affine constraints (2.2) and a time interval 𝐼⊂𝐑, a point 𝑞∈𝑄 such that 𝐴(𝑡,𝑞)≠0,forall𝑡∈𝐼, holds is called a rheonomous affine regular point. On the other hand, a point 𝑞∈𝑄 such that 𝐴(𝑡,𝑞)=0,forall𝑡∈𝐼, holds is called a rheonomous affine equilibrium point.

Note that from Definition 2.2, each point in 𝑄 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 𝐴-rheonomous affine constraints, which are originally obtained in the first paper [38]. From (2.3) in Assumption 2.1, the 𝑛−𝑚 row vectors of 𝐵(𝑞) in the 𝐴-rheonomous affine constraints (2.2) are independent of each other at any point 𝑞∈𝑄. So, we consider 𝑚 vectors which are independent of each other and annihilators of the 𝑛−𝑚 row vectors of 𝐵(𝑞), and denote them by 𝑌₁,…,𝑌_𝑚 as vector fields on 𝑄. Moreover, we also denote a space spanned by 𝑌₁,…,𝑌_𝑚, that is, a distribution on 𝑄 by 𝐷∶=span𝑌₁,…,𝑌_𝑚. (2.4) Since the basial vectors of 𝐷: 𝑌₁,…,𝑌_𝑚 are independent of each other at any point 𝑞∈𝑄, 𝐷 is a nonsingular distribution, that is, dim𝐷𝑞=𝑚,∀𝑞∈𝑄, (2.5) holds. A curve 𝑞∶𝐼→𝑄 is said to be satisfied the 𝐴-rheonomous affine constraints (2.2) if for a vector field on 𝑄: 𝑋 and the generalized velocity of 𝑞: 𝑞̇∈𝑇_𝑞(𝑡)𝑄: 𝑞̇(𝑡)−𝑋𝑡,𝑞(𝑡)∈𝐷𝑞(𝑡),∀𝑡∈𝐼, (2.6) holds as shown in Figure 3. We call this a rheonomous affine vector field.

Figure 3 

Geometric representation of 𝐴-rheonomous affine constraints.

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

Proposition 2.3 (see [38]). For the 𝐴-rheonomous affine constraints (2.2), the component of the rheonomous affine vector field 𝑋, and a time interval 𝐼⊂𝐑: 𝐴𝑡,𝑞+𝐵𝑞𝑋𝑡,𝑞=0,∀𝑞∈𝑄,∀𝑡∈𝐼, (2.7) holds.

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

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

In order to derive some theoretical analysis on 𝐴-rheonomous affine constraints (2.2), the geometric representation of them is quite important. For the geometric representation of the 𝐴-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 𝐴-rheonomous affine constraints (2.2): (𝐷,𝑋) and a time interval 𝐼⊂𝐑, a point 𝑞∈𝑄 is a rheonomous equilibrium point if and only if 𝑋𝑡,𝑞(𝑡)∈𝐷𝑞(𝑡),∀𝑡∈𝐼 (2.8) holds. On the other hand, a point 𝑞∈𝑄 is a rheonomous regular point if and only if 𝑋𝑡,𝑞(𝑡)∉𝐷𝑞(𝑡),∀𝑡∈𝐼 (2.9) 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 𝑍,𝑊 is defined as an operator defined on 𝑇𝑄: [𝑍,𝑊]∶=𝜕𝑊𝜕𝑞𝑍−𝜕𝑍𝜕𝑞𝑊. (2.10) Based on the Lie bracket above, the rheonomous bracket is defined as follows.

Definition 2.6 (see [38]). For the vector fields defined on 𝑄 of the geometric representation of the 𝐴-rheonomous affine constraints (2.2): 𝑋,𝑌₁,…,𝑌_𝑚, the rheonomous bracket is an operator: ⟨⋅,⋅⟩: 𝑇𝑄×𝑇𝑄→𝑇𝑄 that satisfies the next three properties.(a)For a rheonomous affine vector field 𝑋, ⟨𝑋,𝑋⟩=0 (2.11) holds.(b)𝐷₀ is defined as a set of vector fields that consists of 𝑌₁,…,𝑌_𝑚 and iterated rheonomous brackets of 𝑋,𝑌₁,…,𝑌_𝑚 and does not contain 𝑋. For a rheonomous affine vector field 𝑋 and a vector field 𝑍∈𝐷₀, ⟨𝑋,𝑍⟩=𝜕𝑍+[]𝜕𝑡𝑋,𝑍,𝑍∈𝐷₀,⟨𝑍,𝑋⟩=−𝜕𝑍+[]𝜕𝑡𝑍,𝑋,𝑍∈𝐷₀ (2.12) hold.(c)For two vector fields 𝑍,𝑊∈𝐷₀, ⟨𝑍,𝑍⟩∶=0,𝑍∈𝐷₀,⟨𝑍,𝑊⟩∶=[𝑍,𝑊],𝑍,𝑊∈𝐷₀ (2.13) hold.

Note that, in Definition 2.6, the rheonomous affine vector field 𝑋 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 𝐴-rheonomous affine constraints (2.2): 𝑋,𝑌₁,…,𝑌_𝑚 and the set of iterated vector fields of them: 𝐷₀, the following properties (a), (b), and (c) hold.(a)Bilinearlity: ⟨⟩⟨⟩⟨⟩,⟨⟩⟨⟩⟨⟩𝑋,𝑎𝑍+𝑏𝑊=𝑎𝑋,𝑍+𝑏𝑋,𝑊𝑎𝑍+𝑏𝑊,𝑋=𝑎𝑍,𝑋+𝑏𝑊,𝑋,𝑍,𝑊∈𝐷₀. (2.14)(b)Skew-symmetry: ⟨𝑋,𝑍⟩=−⟨𝑍,𝑋⟩,𝑍,𝑊∈𝐷₀. (2.15)(c)Jacobi’s identity: ⟨⟨𝑋,𝑍⟩,𝑊⟩+⟨⟨𝑍,𝑊⟩,𝑋⟩+⟨⟨𝑊,𝑋⟩,𝑍⟩=0,𝑍,𝑊∈𝐷₀. (2.16)

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 𝐴-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 𝐶₀(𝑡,𝑞) which contains 𝑌₁,…,𝑌_𝑚 and iterated rheonomous brackets of them, and satisfies ⟨𝑋,𝑊⟩∈𝐶₀, for all 𝑊∈𝐶₀, that is to say, 𝐶₀ is spanned by all the rheonomous brackets of 𝑋,𝑌₁,…,𝑌_𝑚 with the exception of 𝑋. In addition, we also define a smallest and involutive time-varying distribution 𝐶(𝑡,𝑞) which contains all the iterated rheonomous brackets of 𝑋,𝑌₁,…,𝑌_𝑚. We here show necessary and sufficient conditions on complete integrability for the 𝐴-rheonomous affine constraints (2.2), which are originally derived in the first paper [38].

Theorem 3.1 (see [38]). For the 𝐴-rheonomous affine constraints (2.2) defined on an 𝑛-dimensional manifold 𝑄 and a time interval 𝐼⊂𝐑, the following statements (a)–(c) are equivalent to each other. If they hold, the 𝐴-rheonomous affine constraints (2.2) are said to be completely integrable or completely holonomic.(a)There exist 𝑛−𝑚 independent first integrals of the 𝐴-rheonomous affine constraints (2.2): ℎ₁(𝑡,𝑞),…,ℎ_𝑛−𝑚(𝑡,𝑞) such that 𝜕ℎ_𝑖𝑡,𝑞ℎ𝜕𝑡+𝑋𝑡,𝑞_𝑖𝑍ℎ𝑡,𝑞=0,𝑖=1,…,𝑛−𝑚,∀𝑞∈𝑄,∀𝑡∈𝐼,𝑡,𝑞_𝑖𝑡,𝑞=0,∀𝑍∈𝐷,𝑖=1,…,𝑛−𝑚,∀𝑞∈𝑄,∀𝑡∈𝐼,𝑑ℎ₁∧⋯∧𝑑ℎ_𝑛−𝑚𝑡,𝑞≠0,∀𝑞∈𝑄,∀𝑡∈𝐼, (3.1) hold, where 𝑑 is the exterior differential operator defined on 𝑄.(b)There exists an 𝑚-dimensional time-varying integral manifold 𝑆^𝑚(𝑡,𝑞⁰) of the 𝐴-rheonomous affine constraints (2.2) determined by an initial point 𝑞⁰∈𝑄 such that 𝑇_𝑞𝑆^𝑚𝑡,𝑞⁰=𝐷𝑡,𝑞,∀𝑞∈𝑆^𝑚𝑡,𝑞⁰,∀𝑡∈𝐼, (3.2) holds.(c)For a smallest and involutive time-varying distribution 𝐶₀, dim𝐶₀𝑡,𝑞=𝑚,∀𝑞∈𝑄,∀𝑡∈𝐼, (3.3) holds.

From the statement (b) in Theorem 3.1, we can see that there exists an 𝑚-dimensional time-varying integral manifold in the 𝑛-dimensional configuration manifold 𝑄. It is expected that this time-varying integral manifold creates a foliation structure in 𝑄. 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 𝐴-rheonomous affine constraints (2.2) are completely integrable. Onedenotes a set of rheonomous affine regular points by 𝑈^𝑚_𝑎∶=𝑞∈𝑄∣𝐴𝑡,𝑞≠0,∀𝑡∈𝐼, (3.4) and a set of rheonomous affine equilibrium points by 𝑈^𝑚_𝑏∶=𝑞∈𝑄∣𝐴𝑡,𝑞=0,∀𝑡∈𝐼, (3.5) where 𝐼⊂𝐑is a time interval. Then, the following statements (a) and (b) hold.(a)For 𝑈^𝑚_𝑎 and 𝑈^𝑚_𝑏, 𝑈^𝑚_𝑎∩𝑈^𝑚_𝑏=𝜙,𝑈^𝑚_𝑎∪𝑈^𝑚_𝑏=𝑄 (3.6) hold, where 𝜙 denotes the empty set.(b)For a neighborhood 𝑉^𝑚_𝑏⊂𝑈^𝑚_𝑏 of a rheonomous affine equilibrium point 𝑞^𝑒∈𝑈^𝑚_𝑏, dim𝑉^𝑚_𝑏≥𝑚 (3.7) holds.

Proof. Since each point in 𝑄 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 𝑞^𝑒∈𝑈^𝑚_𝑏 satisfies from Proposition 2.5, we can represent 𝑋(𝑡,𝑞^𝑒) as 𝑋𝑡,𝑞^𝑒=𝛽₁𝑡,𝑞^𝑒𝑌₁𝑞^𝑒+⋯+𝛽_𝑚𝑡,𝑞^𝑒𝑌_𝑚𝑞^𝑒, (3.8) with 𝑚 functions 𝛽₁(𝑡,𝑞),…,𝛽_𝑚(𝑡,𝑞). Consider 𝑛−𝑚 vector fields 𝑌_𝑚+1,…,𝑌_𝑛 which are independent of each other and are orthogonal to 𝑌₁,…,𝑌_𝑚. Calculating inner products of 𝑋(𝑡,𝑞^𝑒) and 𝑌_𝑚+1,…,𝑌_𝑛, we have 𝑌_𝑚+1𝑞^𝑒,𝑋𝑡,𝑞^𝑒=⋯=𝑌_𝑛𝑞^𝑒,𝑋𝑡,𝑞^𝑒=0, (3.9) where ⟨⟨⋅,⋅⟩⟩ is the inner product operator of two vector fields. Therefore, since (3.9) means 𝑛−𝑚 equations for the point 𝑞^𝑒, it can be confirmed that the dimension of a neighborhood of 𝑞^𝑒: 𝑉^𝑚_𝑏 is greater than or equal to 𝑚 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 𝑄 for the completely integrable 𝐴-rheonomous affine constraints.

Theorem 3.3. Assume that the 𝐴-rheonomous affine constraints (2.2) are completely integrable. Then, the following statements (a) and (b) hold for a time interval 𝐼⊂𝐑.(a)For a neighborhood of a rheonomous regular point 𝑞⁰∈𝑈^𝑚_𝑎: 𝑉^𝑚_𝑎⊂𝑈^𝑚_𝑎, there exist a local coordinate (𝑈;𝑥₁,…,𝑥_𝑛) and an 𝑚-dimensional time-varying foliation structure in 𝑉^𝑚_𝑎 which is constructed by an 𝑚-dimensional time-varying integral manifold determined by the time variable 𝑡 and the point 𝑞⁰: 𝑆^𝑚𝑡,𝑞⁰∶=𝑞∈𝑉^𝑚_𝑎∣𝑥_𝑚+1𝑞=𝑥_𝑚+1𝜙^𝑋_𝑡𝑞⁰,…,𝑥_𝑛𝑞=𝑥_𝑛𝜙^𝑋_𝑡𝑞⁰, (3.10) where 𝜙^𝑋_𝑡∶𝑄→𝑄 is a flow (one-parameter local transformation group) of 𝑋. In addition, 𝐶₀𝑡,𝑞=𝐷𝑞,𝐶𝑡,𝑞=𝐷𝑞+span{𝑋}𝑡,𝑞,∀𝑞∈𝑈^𝑚_𝑎,∀𝑡∈𝐼,dim𝐶₀𝑡,𝑞=𝑚,dim𝐶𝑡,𝑞=𝑚+1,∀𝑞∈𝑈^𝑚_𝑎,∀𝑡∈𝐼, (3.11)(3.12) hold.(b)For a neighborhood of a rheonomous equilibrium point 𝑞⁰∈𝑈^𝑚_𝑎: 𝑉^𝑚_𝑏⊂𝑈^𝑚_𝑏, there exist a local coordinate (𝑈;𝑥₁,…,𝑥_𝑛) and an 𝑚-dimensional time-invariant foliation structure in 𝑉^𝑚_𝑏 which is constructed by an 𝑚-dimensional time-invariant integral manifold determined by the point 𝑞⁰: 𝑆^𝑚𝑞⁰∶=𝑞∈𝑉^𝑚_𝑏∣𝑥_𝑚+1𝑞=𝑥_𝑚+1𝑞⁰,…,𝑥_𝑛𝑞=𝑥_𝑛𝑞⁰. (3.13) In addition, 𝐶₀𝑡,𝑞=𝐶𝑡,𝑞=𝐷𝑞,∀𝑞∈𝑈^𝑚_𝑏,∀𝑡∈𝐼,dim𝐶₀𝑡,𝑞=dim𝐶𝑡,𝑞=dim𝐷𝑞,∀𝑞∈𝑈^𝑚_𝑏,∀𝑡∈𝐼, (3.14)(3.15) 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 (𝑛+1)-dimensional expanded configuration manifold which is constructed by the product manifold of the space of the time variable 𝐑and the configuration manifold 𝑄 by 𝑄∶=𝐑×𝑄 with the coordinate 𝑞∶=[𝑡𝑞^⊤]^⊤. The 𝐴-rheonomous affine constraints (2.2) defined on 𝑄 can be represented by an expanded Pfaffian equation on 𝑄: Ω𝑞=0 (3.16) with 𝑛−𝑚 expanded differential forms: Ω𝑞∶=𝐴𝑡,𝑞𝑑𝑡+𝐵𝑞𝑑𝑞. (3.17) We can see that 𝑚+1 annihilators of the 𝑛−𝑚 expanded differential forms (3.17) as 𝜕𝑋=𝜕𝑡⊕𝑋,𝑌_𝑖=0⊕𝑌_𝑖,𝑖=1,…𝑚. (3.18) We also consider a diffeomorphism which is a projection from 𝑄 to 𝑄 at the time 𝑡: 𝜋_𝑡∶(𝑡,𝑞)↦𝑞(𝑡), and its derivation: 𝑑𝜋_𝑡∶𝑇_𝑞(𝑡)𝑄→𝑇_𝑞𝑄. Since 𝑑𝜋_𝑡(𝑋(𝑞))=𝑋(𝑞) holds from the definition of 𝑋 in(3.18), we obtain 𝜋_𝑡(𝜙^𝑋_𝑡(𝑞⁰))=𝜙^𝑋_𝑡(𝑞⁰). In addition, from 𝜙^𝑋_𝑡(𝑞⁰)∈𝑆^𝑚+1(𝑞⁰), it turns out that 𝜙^𝑋_𝑡(𝑞⁰)∈𝑆^𝑚(𝑡,𝑞⁰). Thus, by using a local coordinate (𝑈;𝑥₁,…,𝑥_𝑛), we can represent the 𝑚-dimensional time-varying integral manifold for the 𝐴-rheonomous affine constraints (2.2): 𝑆^𝑚(𝑡,𝑞⁰) as (3.10). Furthermore, from the definitions of 𝐶₀ and 𝐶, it can be confirmed that (3.11) and (3.12) holds.

Figure 4 

An illustration of 𝑄 and 𝑄 (𝑛=2,𝑚=1).

Next, we consider the proof of the statement (b). Since 𝑑𝜋_𝑡(𝑋(𝑞))=𝑋(𝑞)∈𝐷(𝑞) holds for a rheonomous affine equilibrium point 𝑞∈𝑉^𝑚_𝑏, we can show 𝜙^𝑋_𝑡(𝑞⁰)∈𝑆^𝑚(𝑞⁰) by setting 𝜋₀(𝑆^𝑚+1(𝑞⁰))=∶𝑆^𝑚(𝑞⁰). Hence, 𝜋_𝑡(𝑆^𝑚+1(𝑞⁰))=𝑆^𝑚(𝑞⁰) holds and this yields 𝑞⁰∈𝑆^𝑚(𝑞⁰). Since the dimension of 𝑉^𝑚_𝑏 is greater than or equal to 𝑚 from Lemma 3.2, the 𝑚-dimensional time-invariant integral manifold for the 𝐴-rheonomous affine constraints (2.2): 𝑆^𝑚(𝑞⁰) can be represented as (3.13). Moreover, 𝑋(𝑡,𝑞)∈𝐷(𝑞)forall𝑡∈𝐼 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 𝑛=3,𝑚=1, respectively. As shown in Figure 5(a), 𝑉^𝑚_𝑎 is divided into a 1-dimensional time-varying foliation structure, and a trajectory starting from an initial point 𝑞₀ at the time 𝑡 is constrained in the foliation 𝑆^𝑚(𝑡,𝑞⁰). However, since the foliation 𝑆^𝑚(𝑡,𝑞⁰) slides as time goes by, the trajectory can move as a 2-dimensional behavior in 𝑉^𝑚_𝑎. In addition, just like the illustration of the tangent space 𝑇_𝑞𝑄 shown in Figure 5(b), ⟨𝑋,𝑌⟩(𝑡,𝑞) lies on 𝐷(𝑞), however, 𝑋(𝑡,𝑞) does not belong to 𝐷(𝑞). This fact also causes the time-varying foliation structure.

(a) Foliation structure

(b) Tangent space

Figure 5 

An illustration for Theorem 3.3(a) (𝑛=3,𝑚=1).

(a) Foliation structure

(b) Tangent space

Figure 6 

An illustration for Theorem 3.3(b) (𝑛=3,𝑚=1).

On the other hand, 𝑉^𝑚_𝑏 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 𝑞₀ at the time 𝑡 is constrained in the foliation 𝑆^𝑚(𝑞⁰) and cannot move to another foliation. Moreover, as the illustration of the tangent space 𝑇_𝑞𝑄 shown in Figure 6(b), not only ⟨𝑋,𝑌⟩(𝑡,𝑞) but also 𝑋(𝑡,𝑞) lies on 𝐷(𝑞), 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 𝐴-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: 𝑞𝑄=𝑞=𝑞¹₂𝑞₃∈𝐑³∣𝑞₂>0,𝑞₃>0 (3.19) with 𝑛=3, and 𝐴-rheonomous affine constraints on 𝑄: 𝑡𝑞₁𝑞₂𝑞_3𝐴(0_𝑡,𝑞)+𝑞₂𝑞₃𝑞₁𝑞₃𝑞₁𝑞₂010_𝐵(𝑞)𝑞̇₁𝑞̇₂𝑞̇₃=0 (3.20) with 𝑚=1. It turns out that Assumption 2.1 holds for (3.20). One of the geometric representation for (3.20) can be obtained as follows: 𝑋𝑡,𝑞=−𝑡𝑞₁00,𝑌𝑞=−𝑞₁0𝑞₃. (3.21) Calculating an iterated rheonomous brackets for 𝑋 and 𝑌 above, we obtain ⟨𝑋,𝑌⟩=𝜕𝑌+[]𝜕𝑡𝑋,𝑌=0+𝜕𝑌𝜕𝑞𝑋−𝜕𝑋𝜕𝑞𝑌=𝑡𝑞₁−𝑡𝑞₁00=0. (3.22) Hence, it turns out that all the iterated rheonomous bracket of 𝑋, 𝑌 are 0. Therefore, we have 𝐶₀=span{𝑌}, (3.23) and it turns out that dim𝐶₀=1,∀𝑞∈𝑄,𝑡∈𝐑, (3.24) holds. From Theorem 3.1, we can see that the 𝐴-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 ℎ₁𝑡,𝑞=𝑞₂−𝑞⁰₂,ℎ₂=𝑞𝑡,𝑞¹𝑞₃𝑒^(1/2)𝑡2𝑞⁰₃−𝑞⁰₁, (3.25) where 𝑞⁰=[𝑞⁰₁𝑞⁰₂𝑞⁰₃]^⊤∈𝑄 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 𝑈^𝑚_𝑎=𝑉^𝑚_𝑎={𝑞∈𝑄|𝑡>0,𝑞₁≠0}, and the set of the rheonomous affine regular points is represented by 𝑈^𝑚_𝑏=𝑉^𝑚_𝑏={𝑞∈𝑄|𝑡>0,𝑞₁=0}. That is to say, the 𝑞₂-𝑞₃ plane is coincident with 𝑉^𝑚_𝑏 and the total space of 𝑄 except the 𝑞₂-𝑞₃ plane is 𝑉^𝑚_𝑎. 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 𝑞⁰∈𝑉^𝑚_𝑎 is included in the 1-dimensional time-varying integral manifold 𝑆^𝑚(𝑡,𝑞⁰) at the time 𝑡, and moves in the 2-dimensional space determined by 𝑞₂=𝑞⁰₂. Therefore, 𝑉^𝑚_𝑎 has a 1-dimensional time-varying foliation structure. Meanwhile, as shown in Figure 7(b), a trajectory starting from a rheonomous affine equilibrium point 𝑞⁰∈𝑉^𝑚_𝑏 is included in the 1-dimensional time-invariant integral manifold 𝑆^𝑚(𝑞⁰) and moves only in 𝑆^𝑚(𝑞⁰). Consequently, 𝑉^𝑚_𝑏 has a 1-dimensional time-invariant foliation structure.

(a) 𝑉𝑚𝑎

(b) 𝑉𝑚𝑏

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 𝐴-rheonomous affine constraints case in this subsection. Now, necessary and sufficient conditions on partial integrability for the 𝐴-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 𝐴-rheonomous affine constraints (2.2) defined on an 𝑛-dimensional manifold 𝑄 and a time interval 𝐼⊂𝐑, the following statements (a)–(c) are equivalent to each other. If they hold, the 𝐴-rheonomous affine constraints (2.2) are said to be 𝑘th-order partially integrable or 𝑘th-order partially holonomic, where 𝑚<𝑘<𝑛.(a)There exist 𝑛−𝑘 independent first integrals of the 𝐴-rheonomous affine constraints (2.2): ℎ₁(𝑡,𝑞),…,ℎ_𝑛−𝑘(𝑡,𝑞) such that 𝜕ℎ_𝑖𝑡,𝑞ℎ𝜕𝑡+𝑋𝑡,𝑞_𝑖𝑍ℎ𝑡,𝑞=0,𝑖=1,…,𝑛−𝑘,∀𝑞∈𝑄,∀𝑡∈𝐼,𝑡,𝑞_𝑖𝑡,𝑞=0,∀𝑍∈𝐶₀,𝑖=1,…,𝑛−𝑘,∀𝑞∈𝑄,∀𝑡∈𝐼,𝑑ℎ₁∧⋯∧𝑑ℎ_𝑛−𝑘𝑡,𝑞≠0,∀𝑞∈𝑄,∀𝑡∈𝐼, (3.26) hold(b)There exists a 𝑘-dimensional time-varying integral manifold 𝑆^𝑘(𝑡,𝑞⁰) of the 𝐴-rheonomous affine constraints (2.2) determined by an initial point 𝑞⁰∈𝑄 such that 𝑇_𝑞𝑆^𝑘𝑡,𝑞⁰=𝐶₀𝑡,𝑞,∀𝑞∈𝑆^𝑘𝑡,𝑞⁰,∀𝑡∈𝐼, (3.27) holds.(c)For a smallest and involutive time-varying distribution 𝐶₀, dim𝐶₀𝑡,𝑞=𝑘,∀𝑞∈𝑄,∀𝑡∈𝐼, (3.28) holds.

From the statement (b) in Theorem 3.5, it turns out that there exists a 𝑘(𝑚<𝑘<𝑛)-dimensional time-varying integral manifold in the 𝑛-dimensional configuration manifold 𝑄. Similar to the results in the previous section, we can expect that this time-varying integral manifold creates a foliation structure in 𝑄. 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 𝐴-rheonomous affine constraints (2.2) are 𝑘th-order completely integrable. One calls a point 𝑞∈𝑄 such that 𝑋(𝑡,𝑞)∉𝐶₀(𝑡,𝑞),forall𝑡∈𝐼  a 𝑘th order rheonomous affine regular point, and denotes a set of 𝑘th-order rheonomous affine regular points by 𝑈^𝑘_𝑎∶=𝑞∈𝑄∣𝑋𝑡,𝑞∉𝐶₀𝑡,𝑞,∀𝑡∈𝐼, (3.29) Where 𝐼⊂𝐑is a time interval. On the other hand, one call a point 𝑞∈𝑄 such that 𝑋(𝑡,𝑞)∈𝐶₀(𝑡,𝑞),forall𝑡∈𝐼  a 𝑘th-order rheonomous affine equilibrium point, and denotes a set of 𝑘th-order rheonomous affine equilibrium points by 𝑈^𝑘_𝑏∶=𝑞∈𝑄∣𝑋𝑡,𝑞∉𝐶₀𝑡,𝑞∀𝑡∈𝐼. (3.30) Then, the following statements (a) and (b) hold.(a)For 𝑈^𝑘_𝑎 and 𝑈^𝑘_𝑏, 𝑈^𝑘_𝑎∩𝑈^𝑘_𝑏=𝜙,𝑈^𝑘_𝑎∪𝑈^𝑘_𝑏=𝑄 (3.31) hold.(b)For a neighborhood 𝑉^𝑘_𝑏⊂𝑈^𝑘_𝑏 of a rheonomous affine equilibrium point 𝑞^𝑒∈𝑈^𝑚_𝑏, dim𝑉^𝑘_𝑏≥𝑘 (3.32) holds.

Proof. Similar to the proof of Lemma 3.2, since each point in 𝑄 is classified into either a 𝑘th-order rheonomous affine regular point or a 𝑘th-order rheonomous affine equilibrium point from the definition of these points, the statement (a) obviously holds. Now, consider vector fields 𝑌_𝑚+1,…,𝑌_𝑘 which are independent of 𝑌₁,…,𝑌_𝑚 and satisfy 𝐶₀=span𝑌₁,…,𝑌_𝑚,𝑌_𝑚+1,…,𝑌_𝑘. (3.33) Since a 𝑘th-order rheonomous affine equilibrium point 𝑞^𝑒∈𝑈^𝑚_𝑏 satisfies 𝑋(𝑞^𝑒)∈𝐶₀(𝑞^𝑒), 𝑋(𝑡,𝑞^𝑒) can be represented as 𝑋𝑡,𝑞^𝑒=𝛽₁𝑡,𝑞^𝑒𝑌₁𝑞^𝑒+⋯+𝛽_𝑘𝑡,𝑞^𝑒𝑌_𝑘𝑞^𝑒, (3.34) with 𝑘 functions 𝛽₁(𝑡,𝑞),…,𝛽_𝑘(𝑡,𝑞). We here consider 𝑛−𝑘 vector fields 𝑌_𝑘+1,…,𝑌_𝑛 which are independent of each other and are orthogonal to 𝑌₁,…,𝑌_𝑘. Calculating inner products of 𝑋(𝑡,𝑞^𝑒) and 𝑌_𝑘+1,…,𝑌_𝑛, we have 𝑌_𝑘+1𝑞^𝑒,𝑋𝑡,𝑞^𝑒=⋯=𝑌_𝑛𝑞^𝑒,𝑋𝑡,𝑞^𝑒=0. (3.35) So, since (3.35) can be considered as 𝑛−𝑘 equations for the point 𝑞^𝑒, we can see that the dimension of a neighborhood of 𝑞^𝑒: 𝑉^𝑘_𝑏 is greater than or equal to 𝑘 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 𝑄 for the partially integrable 𝐴-rheonomous affine constraints as follows.

Theorem 3.7. Assume that the 𝐴-rheonomous affine constraints (2.2) are 𝑘th order partially integrable. Then, the following statements (a) and (b) hold for a time interval 𝐼⊂𝐑.(a)For a neighborhood of a 𝑘th-order rheonomous regular point 𝑞⁰∈𝑈^𝑘_𝑎: 𝑉^𝑘_𝑎⊂𝑈^𝑘_𝑎, there exist a local coordinate (𝑈;𝑥₁,…,𝑥_𝑛) and a 𝑘-dimensional time-varying foliation structure in 𝑉^𝑘_𝑎 which is constructed by a 𝑘-dimensional time-varying integral manifold determined by the time variable 𝑡 and the point 𝑞⁰: 𝑆^𝑘𝑡,𝑞⁰∶=𝑞∈𝑉^𝑘_𝑎∣𝑥_𝑘+1𝑞=𝑥_𝑘+1𝜙^𝑋_𝑡𝑞⁰,…,𝑥_𝑛𝑞=𝑥_𝑛𝜙^𝑋_𝑡𝑞⁰. (3.36) In addition, 𝐶𝑡,𝑞=𝐶₀𝑡,𝑞+span{𝑋}𝑡,𝑞,∀𝑞∈𝑈^𝑘_𝑎,∀𝑡∈𝐼,dim𝐶₀𝑡,𝑞=𝑘,dim𝐶𝑡,𝑞=𝑘+1,∀𝑞∈𝑈^𝑘_𝑎,∀𝑡∈𝐼, (3.37) hold.(b)For a neighborhood of a 𝑘th-order rheonomous equilibrium point 𝑞⁰∈𝑈^𝑘_𝑎: 𝑉^𝑚_𝑏⊂𝑈^𝑘_𝑏, there exist a local coordinate (𝑈;𝑥₁,…,𝑥_𝑛) and a 𝑘-dimensional time-invariant foliation structure in 𝑉^𝑘_𝑏 which is constructed by a 𝑘-dimensional time-invariant integral manifold determined by the point 𝑞⁰: 𝑆^𝑘𝑞⁰∶=𝑞∈𝑉^𝑚_𝑏∣𝑥_𝑘+1𝑞=𝑥_𝑘+1𝑞⁰,…,𝑥_𝑛𝑞=𝑥_𝑛𝑞⁰. (3.38) In addition, 𝐶₀𝑡,𝑞=𝐶𝑡,𝑞,∀𝑞∈𝑈^𝑘_𝑏,∀𝑡∈𝐼,dim𝐶₀𝑡,𝑞=dim𝐶𝑡,𝑞=𝑘,∀𝑞∈𝑈^𝑘_𝑏,∀𝑡∈𝐼, (3.39) 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 𝑛=3, 𝑚=1, 𝑘=2, respectively. As shown in Figure 8(a), 𝑉^𝑘_𝑎 is divided into a 2-dimensional time-varying foliation structure, and a trajectory that starts from an initial point 𝑞₀ at the time 𝑡 is included in the foliation 𝑆^𝑘(𝑡,𝑞⁰). However, since the foliation 𝑆^𝑘(𝑡,𝑞⁰) slides as time goes by, the trajectory can move as a 3-dimensional behavior in 𝑉^𝑚_𝑎. In addition, like the illustration of the tangent space 𝑇_𝑞𝑄 shown in Figure 8(b), ⟨𝑋,𝑌⟩(𝑡,𝑞) lies on 𝐶(𝑡,𝑞), however, 𝑋(𝑡,𝑞) does not belong to 𝐶(𝑡,𝑞). This result also causes the time-varying foliation structure.

(a) Foliation structure

(b) Tangent space

Figure 8 

An illustration for Theorem 3.7(a) (𝑛=3,𝑚=1,𝑘=2).

(a) Foliation structure

(b) Tangent space

Figure 9 

An illustration for Theorem 3.7(b) (𝑛=3,𝑚=1,𝑘=2).

In contrast, 𝑉^𝑘_𝑏 is divided into a 2-dimensional time-invariant foliation structure as shown in Figure 9(a), Therefore, a trajectory starting from an initial point 𝑞₀ at the time 𝑡 is constrained in the foliation 𝑆^𝑚(𝑞⁰) and cannot move to another foliation. Moreover, as the illustration of the tangent space 𝑇_𝑞𝑄 shown in Figure 9(b), 𝑋(𝑡,𝑞) as well as ⟨𝑋,𝑌⟩(𝑡,𝑞) lies on 𝐶(𝑡,𝑞), 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 𝐴-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: 𝑞𝑄=𝑞=𝑞¹₂𝑞₃∈𝐑³∣𝑞₂>0,𝑞₃>0 (3.40) with 𝑛=3 and 𝐴-rheonomous affine constraints on 𝑄: 𝑡𝑞₁𝑞₂𝑞₃𝑞₂𝑞_{3𝐴(𝑡,𝑞)}+𝑞₂𝑞₃𝑞₁𝑞₃𝑞₁𝑞₂01𝑞_2𝐵(𝑞)𝑞̇₁𝑞̇₂𝑞̇₃=0 (3.41) with 𝑚=1. We can see that Assumption 2.1 holds for (3.41). One of the geometric representation for (3.41) can be obtained as follows: 𝑋𝑡,𝑞=−𝑡𝑞₁+𝑞₁𝑞₃−𝑞₂𝑞₃0,𝑌𝑞=−𝑞₁+𝑞₁𝑞₃−𝑞₂𝑞₃𝑞₃. (3.42) By calculating iterated rheonomous brackets for 𝑋 and 𝑌 above, we have ⟨𝑋,𝑌⟩=𝜕𝑌+[]𝜕𝑡𝑋,𝑌=0+𝜕𝑌𝜕𝑞𝑋−𝜕𝑋𝜕𝑞𝑌=−𝑞₁𝑞₃𝑞₂𝑞₃0,⟨⟨=𝜕⟨⟩𝑋,𝑋,𝑌⟩⟩𝑋,𝑌+[⟨=𝜕𝑡𝑋,𝑋,𝑌⟩]𝑡𝑞₁𝑞₃−𝑞₁𝑞²₃−𝑞₂𝑞²₃0−𝑡𝑞₁𝑞₃−𝑞₁𝑞²₃−𝑞₂𝑞²₃0⟨⟨=[⟨==0,𝑌,𝑋,𝑌⟩⟩𝑌,𝑋,𝑌⟩]−𝑞₁𝑞²₃−𝑞₂𝑞²₃+𝑞₂𝑞₃0−𝑞₁𝑞₃−𝑞₁𝑞²₃−𝑞₂𝑞²₃0=−𝑞₁𝑞₃𝑞₂𝑞₃0,⋮ (3.43) Therefore, we have 𝐶₀=span{𝑌,⟨𝑋,𝑌⟩}, (3.44) and we can see that dim𝐶₀=2,∀𝑞∈𝑄,∀𝑡∈𝐑 (3.45) holds. From Theorem 3.3, we can find that the 𝐴-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 ℎ₁=𝑞𝑡,𝑞¹𝑞₂𝑞₃𝑒^(1/2)𝑡2𝑞⁰₂𝑞⁰₃−𝑞⁰₁, (3.46) where 𝑞⁰=[𝑞⁰₁𝑞⁰₂𝑞⁰₃]^⊤∈𝑄 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 𝑈^𝑘_𝑎=𝑉^𝑘_𝑎={𝑞∈𝑄|𝑡>0,𝑞₁≠0}, and the set of the 2nd-order rheonomous affine regular points is given by 𝑈^𝑚_𝑏=𝑉^𝑚_𝑏={𝑞∈𝑄|𝑡>0,𝑞₁=0}. Hence, the 𝑞₂-𝑞₃ plane corresponds to 𝑉^𝑘_𝑏 and the total space of 𝑄 except the 𝑞₂-𝑞₃ plane is 𝑉^𝑘_𝑎. 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 𝑞⁰∈𝑉^𝑘_𝑎 is included in the 2-dimensional time-varying integral manifold 𝑆^𝑘(𝑡,𝑞⁰) at the time 𝑡, and it moves in the 3-dimensional space determined by 𝑞₂=𝑞⁰₂. Therefore, 𝑉^𝑘_𝑎 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 𝑞⁰∈𝑉^𝑚_𝑏 is included in the 2-dimensional time-invariant integral manifold 𝑆^𝑘(𝑞⁰) and moves only in 𝑆^𝑘(𝑞⁰). Consequently, 𝑉^𝑘_𝑏 has a 2-dimensional time-invariant foliation structure.

(a) 𝑉𝑘𝑎

(b) 𝑉𝑘𝑏

Figure 10 

Foliation structures of Example 3.8.

3.3. Completely Nonintegrable Case

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

Theorem 3.9 (see [38]). For the 𝐴-rheonomous affine constraints (2.2) defined on an 𝑛-dimensional manifold 𝑄 and a time interval 𝐼⊂𝐑, the following statements (a), (b) are equivalent to each other. If they hold, the 𝐴-rheonomous affine constraints (2.2) are said to be completely nonintegrable or completely nonholonomic.(a)There do not exist independent first integrals of the 𝐴-rheonomous affine constraints (2.2).(b)For a smallest and involutive time-varying distribution 𝐶₀, dim𝐶₀𝑡,𝑞=𝑛,∀𝑞∈𝑄,∀𝑡∈𝐼, (3.47) holds.

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

Theorem 3.10. Assume that the 𝐴-rheonomous affine constraints (2.2) are completely nonintegrable. Then, for a time interval 𝐼⊂𝐑, 𝑋𝑡,𝑞⁰∈𝐶₀𝑞⁰,𝑞⁰∈𝑄,𝑡∈𝐼, (3.48) holds and there is no foliation structure in 𝑄. In addition, 𝐶₀𝑡,𝑞=𝐶𝑡,𝑞=𝑇_𝑞𝑄,∀𝑞∈𝑄,∀𝑡∈𝐼,dim𝐶₀𝑡,𝑞=dim𝐶𝑡,𝑞=𝑛,∀𝑞∈𝑄,∀𝑡∈𝐼, (3.49) hold.

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

For Theorem 3.10, the illustration is depicted for the case of 𝑛=3,𝑚=1,𝑘=2 in Figure 11. As shown in Figure 11(a), a trajectory starting from an initial point 𝑞⁰∈𝑄 can move in 𝑄 without constraints by integral manifolds. Therefore, 𝑄 is not divided into a foliation structure. In addition, as the illustration of the tangent space 𝑇_𝑞𝑄 shown in Figure 11(b), neither ⟨𝑋,𝑌⟩(𝑡,𝑞) nor ⟨𝑋,⟨𝑋,𝑌⟩⟩(𝑡,𝑞) do not belong to 𝐷(𝑞) and three vector fields 𝑌(𝑞),⟨𝑋,𝑌⟩(𝑡,𝑞),⟨𝑋,⟨𝑋,𝑌⟩⟩(𝑡,𝑞) span a 3-dimensional space. This result also causes no foliation structure in 𝑄.

(a) Foliation structure

(b) Tangent space

Figure 11 

An illustration for Theorem 3.10 (𝑛=3,𝑚=1).

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: 𝑞𝑄=𝑞=𝑞¹₂𝑞₃∈𝐑³ (3.50) with 𝑛=3, and 𝐴-rheonomous affine constraints defined on 𝑄: 𝑡𝑞₂−𝑞_{1𝐴(𝑡,𝑞)}+10−1011_𝐵(𝑞)𝑞̇₁𝑞̇₂𝑞̇₃=0 (3.51) with 𝑚=1. We can see that Assumption 2.1 holds for (3.51). One of the geometric representation for (3.51) can be obtained: 𝑋𝑡,𝑞=−𝑡𝑞₂𝑞₁0𝑞=11,𝑌−1. (3.52) Calculating iterated rheonomous brackets for 𝑋 and 𝑌 above, we have ⟨𝑋,𝑌⟩=𝜕𝑌+[]𝜕𝑡𝑋,𝑌=0+𝜕𝑌𝜕𝑞𝑋−𝜕𝑋00,⟨⟨=𝜕⟨⟩𝜕𝑞𝑌=−𝑡𝑋,𝑋,𝑌⟩⟩𝑋,𝑌+[⟨=00+0𝑡0=𝑡0,⟩⟩𝜕⟨⟩𝜕𝑡𝑋,𝑋,𝑌⟩]−1−1⟨⟨𝑋,𝑌,𝑋=−𝑋,𝑌+⟩]=1𝑡0,⟨⟨=[⟨⟩⟩=⟩]⋮𝜕𝑡[⟨𝑋,𝑌,𝑋𝑌,𝑋,𝑌⟩⟩𝑌,𝑋,𝑌⟩]=0,⟨⟨𝑋,𝑌,𝑌[⟨𝑋,𝑌,𝑌=0, (3.53) Therefore, we have 𝐶₀=span{𝑌,⟨𝑋,𝑌⟩,⟨𝑋,⟨𝑋,𝑌⟩⟩}, (3.54) and it can be confirmed that dim𝐶₀=3,∀𝑞∈𝑄,∀𝑡≠0 (3.55) holds. From Theorem 3.9, we can see that the 𝐴-rheonomous affine constraints (3.51) are completely nonintegrable. Therefore, from Theorem 3.10, it can be confirmed that there exists no foliation structure in 𝑄.

4. Integrating Algorithm for Integrable 𝐴-Rheonomous Affine Constraints

4.1. Completely Integrable Case

As seen in the first paper [38] and Section 3 in this second paper, if the 𝐴-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 𝑄. If we consider reduction of the dimension of a given configuration manifold with the 𝐴-rheonomous affine constraints, we need the explicit forms of independent first integrals. For scleronomous linear constraints, that is, 𝐵(𝑞)𝑞̇=0, a method on calculation of independent first integrals of them has been well known [17, 18], and for scleronomous affine constraints, that is, 𝐴(𝑞)+𝐵(𝑞)𝑞̇=0, we have developed algorithms to calculate independent first integrals of them in [25]. However, a method to calculate independent first integrals of given 𝐴-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 𝐴-rheonomous affine constraints (2.2).

In this subsection, we first consider the case where the 𝐴-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 𝑛−𝑚 independent first integrals of the 𝐴-rheonomous affine constraints (2.2). Now, we derive an algorithm to calculate the 𝑛−𝑚 independent first integrals. First of all, we can find 𝑛−𝑚 vector fields 𝑌_𝑚+1,…,𝑌_𝑛 such that span𝑌₁,…,𝑌_𝑚,𝑌_𝑚+1,…,𝑌_𝑛=𝐑^𝑛 (4.1) holds for the vector fields of the geometric representation for the 𝐴-rheonomous affine constraints (2.2): 𝑌₁,…,𝑌_𝑚. Then, let us denote flows (1-parameter local transformation groups) of 𝑋 and 𝑌_𝑖 by 𝜙^𝑋_𝑡 and 𝜑^𝑌𝑖_𝜏𝑖 with parameters 𝑡 and 𝜏_𝑖, respectively. We set an initial point at the initial time 𝑡=0 as 𝑞⁰∈𝑄. We also consider 𝑚+1 vector fields defined on the expanded configuration manifold 𝑄=𝐑×𝑄 (3.18), and then their flows on 𝑄 are obtained as 𝜙^𝑋_𝑡𝑡𝜙∶=_𝑋𝑡,𝜑^𝑌𝑖_𝜏𝑖0𝜑∶=_{𝑌𝑖𝜏𝑖},𝑖=1,…,𝑚. (4.2) Calculating the composite mapping of (𝑛+1) flows (4.15) yields Φ∶=𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘⋯∘𝜑^𝑌𝑛_𝜏𝑛=𝑡𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘⋯∘𝜑^𝑌𝑛_𝜏𝑛=𝑡Φ_𝑡, (4.3) where Φ_𝑡∶=𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘⋯∘𝜑^𝑌𝑛_𝜏𝑛 (4.4) is the composite mapping of 𝑛+1 flows 𝜙^𝑋_𝑡,𝜑^𝑌𝑖_𝜏𝑖. From (4.16), it turns out that the projection of Φ onto 𝑄 is equivalent to Φ_𝑡. Hence, by applying the idea of the integrating algorithm for scleronomous linear constraints defined on 𝑄 [17, 18] to 𝐑×𝑄 and considering projection of it onto 𝑄, we can derive the following algorithm to calculate 𝑛−𝑚 independent first integrals of completely integrable 𝐴-rheonomous affine constraints.

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

Step 1. Set 𝑚+1 vector fields 𝑋,𝑌₁,…,𝑌_𝑚 of geometric representation for the 𝐴-rheonomous affine constraints (2.2).

Step 2. For 𝑋,𝑌₁,…,𝑌_𝑚, derive linearly independent vector fields 𝑌_𝑚+1,…,𝑌_𝑛 that satisfy (4.14).

Step 3. Calculate flows of 𝑋,𝑌₁,…,𝑌_𝑛: 𝜙^𝑋_𝑡,𝜑^𝑌1_𝜏𝑖,…,𝜑^𝑌𝑛_𝜏𝑖.

Step 4. Combine 𝑛+1 flows 𝜙^𝑋_𝑡,𝜑^𝑌1_𝜏𝑖,…,𝜑^𝑌𝑛_𝜏𝑖 derived in Step 4 as (4.17).

Step 5. Set 𝑞=Φ_𝑡(𝜏) and calculate the inverse function 𝜏=Φ_𝑡⁻¹(𝑞), where 𝜏=[𝜏₁⋯𝜏_𝑛]^⊤∈𝐑^𝑛. Then, the last 𝑛−𝑚 components of Φ_𝑡⁻¹ 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 𝐴-rheonomous affine constraints (2.2). Moreover, the existence of the inverse mapping Φ_𝑡⁻¹ which appears at Step 5 in Algorithm 4.1 is guaranteed by the following proposition.

Proposition 4.2. Assume that the 𝐴-rheonomous affine constraints (2.2) are completely integrable. Then, there exists a time interval 𝐼⊂𝐑and Φ_𝑡(𝜏) is a diffeomorphism at any time 𝑡∈𝐼. That is to say, there exists its inverse mapping Φ_𝑡⁻¹.

Proof. First, we set 𝜏∶=[𝑡𝜏^⊤]^⊤. Calculating the partial differential of (4.16) with the chain rule of differential calculation, we have 𝜕Φ=𝜕𝜕𝑡𝜕𝑡𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘⋯∘𝜑^𝑌𝑛_𝜏𝑛=𝑋𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘⋯∘𝜑^𝑌𝑛_𝜏𝑛,𝜕Φ𝜕𝜏_𝑖=𝜕𝜙^𝑋_𝑡𝜕𝜏𝜕𝜑^𝑌1_𝜏1𝜕𝜏⋯𝜕𝜑^𝑌𝑖−1_𝜏𝑖−1𝜕𝜏𝜕𝜕𝜏_𝑖𝜑^𝑌𝑖_𝜏𝑖∘⋯∘𝜑^𝑌𝑛_𝜏𝑛=𝜕𝜙^𝑋_𝑡𝜕𝜏𝜕𝜑^𝑌1_𝜏1𝜕𝜏⋯𝜕𝜑^𝑌𝑖−1_𝜏𝑖−1𝜕𝜏𝑌_𝑖𝜑^𝑌𝑖_𝜏𝑖∘⋯∘𝜑^𝑌𝑛_𝜏𝑛. (4.5) Substituting 𝜏=0 into (4.5), we obtain 𝜕Φ𝜕𝑡_𝜏=0=𝑋𝑞⁰,𝜕Φ𝜕𝜏_𝑖𝜏=0=𝑌_𝑖𝑞⁰, (4.6) that is to say, 𝜕Φ𝜕𝑡_𝜏=0=𝑋𝑞⁰𝑌₁𝑞⁰⋯𝑌_𝑛𝑞⁰. (4.7) Since 𝑋,𝑌₁,…,𝑌_𝑛 are linearly independent of each other, 𝜕rankΦ𝜕𝜏_𝜏=0=𝑛+1 (4.8) holds. Hence it can be confirmed that Φ(𝜏) is a diffeomorphism by the implicit function theorem. Since the projection of Φ(𝜏) onto 𝑄 is equivalent to Φ_𝑡(𝜏), Φ_𝑡(𝜏) 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 𝐴-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 𝑚=1, 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 𝑌₁∶=𝑌. We can easily find two vector fields 𝑌₂,𝑌₃ satisfying (4.14) as 𝑌₂=010,𝑌₃=100, (4.9) and then flows of 𝑋, 𝑌₁,𝑌₂,𝑌₃ can be obtained as 𝜙^𝑋_𝑡=𝑞⁰₁𝑒^{−(1/2)𝑡2}𝑞⁰₂𝑞⁰₃,𝜑^𝑌1_𝜏1=𝑞⁰₁𝑒^−𝜏1𝑞⁰₂𝑞⁰₃𝑒^𝜏1,𝜑^𝑌2_𝜏2=𝑞⁰₁𝜏₂+𝑞⁰₂𝑞⁰₃,𝜑^𝑌3_𝜏3=𝜏₃+𝑞⁰₁𝑞⁰₂𝑞⁰₃, (4.10) where 𝑞⁰=[𝑞⁰₁𝑞⁰₂𝑞⁰₃]^⊤∈𝑄 is an initial point of the flows at the initial time. Combining the flows (4.10) as (4.17), we have Φ_𝑡(𝜏)=𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘𝜑^𝑌2_𝜏2∘𝜑^𝑌3_𝜏3=𝜏₃+𝑞⁰₁𝑒^−𝜏1𝑒^{−(1/2)𝑡2}𝜏₂+𝑞⁰₂𝑞⁰₃𝑒^𝜏1, (4.11) and from the equation 𝑞=Φ_𝑡(𝜏), we can calculate the inverse mapping of Φ_𝑡 as follows: Φ_𝑡⁻¹𝑞=𝑞log⁰³𝑞₃𝑞₂−𝑞⁰₂𝑞₁𝑞₃𝑒^(1/2)𝑡2𝑞⁰₃−𝑞⁰₁. (4.12) Consequently, we have two independent first integrals of (3.50): ℎ₁𝑡,𝑞=𝑞₂−𝑞⁰₂,ℎ₂=𝑞𝑡,𝑞¹𝑞₃𝑒^(1/2)𝑡2𝑞⁰₃−𝑞⁰₁ (4.13) 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 𝐴-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 𝑄. If we consider reduction of the dimension of a given configuration manifold with the 𝐴-rheonomous affine constraints, we need the explicit forms of independent first integrals. For scleronomous linear constraints, that is, 𝐵(𝑞)𝑞̇=0, a method on calculation of independent first integrals of them has been well known [17, 18], and for scleronomous affine constraints, that is, 𝐴(𝑞)+𝐵(𝑞)𝑞̇=0, we have developed algorithms to calculate independent first integrals of them in [25]. However, a method to calculate independent first integrals of given 𝐴-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 𝐴-rheonomous affine constraints (2.2).

In this subsection, we first consider the case where the 𝐴-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 𝑛−𝑚 independent first integrals of the 𝐴-rheonomous affine constraints (2.2). Now, we derive an algorithm to calculate the 𝑛−𝑚 independent first integrals. First of all, we can find 𝑛−𝑚 vector fields 𝑌_𝑚+1,…,𝑌_𝑛 such that span𝑌₁,…,𝑌_𝑚,𝑌_𝑚+1,…,𝑌_𝑛=𝐑^𝑛 (4.14) holds for the vector fields of the geometric representation for the 𝐴-rheonomous affine constraints (2.2): 𝑌₁,…,𝑌_𝑚. Then, let us denote flows (1-parameter local transformation groups) of 𝑋 and 𝑌_𝑖 by 𝜙^𝑋_𝑡 and 𝜑^𝑌𝑖_𝜏𝑖 with parameters 𝑡 and 𝜏_𝑖, respectively. We set an initial point at the initial time 𝑡=0 as 𝑞⁰∈𝑄. We also consider 𝑚+1 vector fields defined on the expanded configuration manifold 𝑄=𝐑×𝑄 (3.18), and then their flows on 𝑄 are obtained as 𝜙^𝑋_𝑡𝑡𝜙∶=_𝑋𝑡,𝜑^𝑌𝑖_𝜏𝑖0𝜑∶=_{𝑌𝑖𝜏𝑖},𝑖=1,…,𝑚. (4.15) Calculating the composite mapping of (𝑛+1) flows (4.15) yields Φ∶=𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘⋯∘𝜑^𝑌𝑛_𝜏𝑛=𝑡𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘⋯∘𝜑^𝑌𝑛_𝜏𝑛=𝑡Φ_𝑡, (4.16) where Φ_𝑡∶=𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘⋯∘𝜑^𝑌𝑛_𝜏𝑛 (4.17) is the composite mapping of 𝑛+1 flows 𝜙^𝑋_𝑡,𝜑^𝑌𝑖_𝜏𝑖. From (4.16), it turns out that the projection of Φ onto 𝑄 is equivalent to Φ_𝑡. Hence, by applying the idea of the integrating algorithm for scleronomous linear constraints defined on 𝑄 [17, 18] to 𝐑×𝑄 and considering projection of it onto 𝑄, we can derive the following algorithm to calculate 𝑛−𝑚 independent first integrals of completely integrable 𝐴-rheonomous affine constraints.

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

Step 1. Set 𝑚+1 vector fields 𝑋,𝑌₁,…,𝑌_𝑚 of geometric representation for the 𝐴-rheonomous affine constraints (2.2).

Step 2. By calculating iterated Lie brackets of 𝑋,𝑌₁,…,𝑌_𝑚, derive 𝑘−𝑚 linearly independent vector fields 𝑌_𝑚+1,…,𝑌_𝑘 that satisfy 𝐶₀=span𝑌₁,…,𝑌_𝑚,𝑌_𝑚+1,…,𝑌_𝑘. (4.18)

Step 3. For 𝑋,𝑌₁,…,𝑌_𝑘, derive linearly independent vector fields 𝑌_𝑘+1,…,𝑌_𝑛 that satisfy span𝑌₁,…,𝑌_𝑘,𝑌_𝑘+1,…,𝑌_𝑛=𝐑^𝑛. (4.19)

Step 4. Calculate flows of 𝑋,𝑌₁,…,𝑌_𝑛: 𝜙^𝑋_𝑡,𝜑^𝑌1_𝜏𝑖,…,𝜑^𝑌𝑛_𝜏𝑖.

Step 5. Combine 𝑛+1 flows derived in Step 4 as (4.17).

Step 6. Set 𝑞=Φ_𝑡(𝜏) and calculate the inverse function 𝜏=Φ_𝑡⁻¹(𝑞). Then, the last 𝑛−𝑘 components of Φ_𝑡⁻¹ 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 𝑘−𝑚 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 Φ_𝑡⁻¹ at Step 5 in Algorithm 4.4 is guaranteed by the following proposition.

Proposition 4.5. Assume that the 𝐴-rheonomous affine constraints (2.2) are 𝑘(𝑚<𝑘<𝑛)th-order partially integrable. Then, there exists a time interval 𝐼⊂𝐑and Φ_𝑡(𝜏) is a diffeomorphism at any time 𝑡∈𝐼. That is, there exists its inverse mapping Φ_𝑡⁻¹.

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 𝐴-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 𝑘=2, 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 𝑌₁∶=𝑌. Calculating iterated rheonomous brackets for (3.42), we have 𝐶₀=span{𝑌₁,⟨𝑋,𝑌₁⟩}, where 𝑋,𝑌₁=−𝑞₁𝑞₃𝑞₂𝑞₃0. (4.20) Next, we set 𝑌₂∶=⟨𝑋,𝑌₁⟩ and calculate 𝑌₃ satisfying (4.19) as 𝑌₃=100 (4.21) and the flows of 𝑋, 𝑌₁, 𝑌₂, 𝑌₃ can be obtained as 𝜙^𝑋_𝑡=𝑞⁰₁𝑒^{−(1/2)𝑡2+𝑞03𝑡}𝑞⁰₂𝑒^{−𝑞03𝑡}𝑞⁰₃,𝜑^𝑌1_𝜏1=𝑞⁰₁𝑒^{𝑞03(𝑒𝜏1−1)−𝜏1}𝑞⁰₂𝑒^{−𝑞03(𝑒𝜏1−1)}𝑞⁰₃𝑒^𝜏1,𝜑^𝑌2_𝜏2=𝑞⁰₁𝑒^{−𝑞03𝜏2}𝑞⁰₂𝑒^𝑞03𝜏2𝑞⁰₃,𝜑^𝑌3_𝜏3=𝜏₃+𝑞⁰₁𝑞⁰₂𝑞⁰₃, (4.22) where 𝑞⁰=[𝑞⁰₁𝑞⁰₂𝑞⁰₃]^⊤∈𝑄 is an initial point of the flows at the initial time. Combining (4.22) as (4.17), we have Φ_𝑡(𝜏)=𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘𝜑^𝑌2_𝜏2∘𝜑^𝑌3_𝜏3=𝜏₃+𝑞⁰₁𝑒^{−𝑞03𝜏2}𝑒^{𝑞03(𝑒𝜏1−1)}𝑒^{−(1/2)𝑡2+𝑞03𝑒𝜏1𝑡}𝑞⁰₂𝑒^𝑞03𝜏2𝑒^{−𝑞03(𝑒𝜏1𝑡)}𝑞⁰₃𝑒^𝜏1, (4.23) and from the equation 𝑞=Φ_𝑡(𝜏), we can calculate its inverse mapping: Φ_𝑡⁻¹𝑞=𝑞log⁰³𝑞₃𝑞₃(𝑡+1)𝑞⁰₃+1𝑞⁰₃𝑞log²𝑞⁰₂𝑞−1₁𝑞₂𝑞₃𝑒^(1/2)𝑡2𝑞⁰₂𝑞⁰₃−𝑞⁰₁. (4.24) Consequently, we can obtain one independent first integral of (3.41): ℎ₁=𝑞𝑡,𝑞¹𝑞₂𝑞₃𝑒^(1/2)𝑡2𝑞⁰₂𝑞⁰₃−𝑞⁰₁ (4.25) 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 𝑥-axis and the 𝑦-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: 𝑥𝑦𝑄=𝑞=∈𝐑²∣𝑦≥0, (5.1) where 𝑛=2. The position of the weight in 𝑄 is represented by (𝑥,𝑦). In addition, the angle of the string in the clockwise direction measured from the 𝑦-axis is denoted by 𝜃. 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 𝐿(𝑡).

Figure 12 

A single pendulum with a time-varying elastic string.

This system is undoubtedly subject to a constraint on the configuration variables as 𝑥²+𝑦²=𝐿(𝑡). (5.2) 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 (𝑥,𝑦) by 𝜃. However, we consider this system as a physical example which is subject to an integrable 𝐴-rheonomous affine constraint. Calculating time derivative of (5.2), we have 𝑥𝑥̇+𝑦𝑦̇=𝐿̇(𝑡)𝐿(𝑡). (5.3) Now, for the sake of simplicity, we set 𝑁(𝑡)∶=𝐿̇(𝑡)𝐿(𝑡). Then, (5.3) can be represented as an 𝐴-rheonomous affine constraint defined on 𝑄: −𝑁(𝑡)_𝐴(𝑡)+𝑥𝑦_𝐵(𝑞)𝑥̇𝑦̇=0, (5.4) where 𝑚=1. Let us start with the above 𝐴-rheonomous affine constraint (5.4). One of geometric representation for the 𝐴-rheonomous affine constraints (5.4) can be derived as 0𝑋=𝑁(𝑡)𝑦1−,𝑌=𝑥𝑦. (5.5) For the vector fields 𝑋,𝑌, we calculate a rheonomous bracket as ⟨𝑋,𝑌⟩=𝜕𝑌+[]=0𝜕𝑡𝑋,𝑌𝑁(𝑡)𝑥𝑦³−0𝑁(𝑡)𝑥𝑦³=00, (5.6) and so we can see that all the iterated rheonomous brackets of 𝑋,𝑌 are 0. Consequently, we obtain 𝐶₀=span{𝑌}, (5.7) and hence dim𝐶₀=1,∀𝑞∈𝑄,∀𝑡∈𝐼∶=𝐑 (5.8) holds. From the condition (c) in Theorem 3.1, it turns out that the 𝐴-rheonomous affine constraint (5.4) is completely integrable, and this fact coincides with the problem setting of this system. Furthermore, the condition (a) in Theorem 3.1, we can see that there exist one first integral of the 𝐴-rheonomous affine constraint (5.4). So, we next calculate it by using the integrating algorithm introduced in Section 4.1 (Algorithm 4.1). We set 𝑌₁∶=𝑌 and a vector field 𝑌₂ that satisfies span{𝑌₁,𝑌₂}=𝐑² as 𝑌₂=10. (5.9) Next, the flows of 𝑋,𝑌₁,𝑌₂ can be calculated as 𝜙^𝑋_𝑡=𝑥⁰𝑦⁰²+𝐿(𝑡)²−𝐿(0)²,𝜑^𝑌1_𝜏1=𝜏₁+𝑥⁰𝜏²₁+𝑦⁰²,𝜑^𝑌2_𝜏2=𝜏₂+𝑥⁰𝑦⁰, (5.10) where 𝑞⁰=[𝑥⁰𝑦⁰]^⊤ is an initial point of the flows. Now, combining the flows (5.10) as (4.16), we then obtain Φ_𝑡=𝜙^𝑋_𝑡∘𝜑^𝑌1_𝜏1∘𝜑^𝑌2_𝜏2=𝜏₁+𝑥⁰𝜏²₁+𝐿(𝑡)²−𝐿(0)²+𝜏₂+𝑦⁰². (5.11) From (5.11), we can calculate the inverse mapping of Φ_𝑡 as Φ_𝑡⁻¹𝑞=𝑥−𝑥⁰𝑥−𝑥⁰²+𝑦²−𝐿(𝑡)²+𝐿(0)²−𝑦⁰. (5.12) Therefore, the first integral can be obtained as the last component of (5.12): 𝑥−𝑥⁰²+𝑦²−𝐿(𝑡)²+𝐿(0)²−𝑦⁰=0. (5.13) For the sake of simplicity, we set the initial values of 𝑥,𝑦,𝐿(𝑡) as 𝑥⁰=𝑦⁰=0,𝐿(0)=0. Thus, from (5.13), we can obtain 𝑥²+𝑦²=𝐿(𝑡)², (5.14) and we can confirm that (5.14) corresponds to the rheonomous holonomic constraint (5.2).

6. Conclusions

In this paper, we have treated the 𝐴-rheonomous affine constraints from the aspects of foliation structures and integrating algorithms. We first investigate foliation structures of configuration manifolds with the 𝐴-rheonomous affine constraints for the three cases: complete integrability, partial integrability, and complete nonintegrability. Next, we have developed two kinds of integrating algorithms that can calculate independent first integrals of the 𝐴-rheonomous affine constraints in the complete integrability and partial integrability cases. Then, a pendulum with a time-varying elastic string as a physical example has been presented in order to show the effectiveness of the new results obtained in this paper.

It has turned out from the results that (a) for the completely and partially integrable 𝐴-rheonomous affine constraints, there exist not only time-invariant foliation structures but also time-varying ones in configuration manifolds, and this fact is similar to the ones for the scleronomous affine constraints [23], (b) for the completely nonintegrable 𝐴-rheonomous affine constraints, there do not exist any foliation structures, (c) independent first integrals of given 𝐴-rheonomous affine constraints can be calculated by using the integrating algorithms, (d) the rheonomous bracket plays important roles in analysis of foliation structures and derivation of integrating algorithms. These results obtained in this paper seem to be very useful for various research fields such as general mathematics and physics as well as control theory.

Throughout the first paper [38] and this second paper, we have considered a new class of constraints, the 𝐴-rheonomous affine constraints, which have been hardly researched so far, and we have developed some fundamental mathematical tools to analyze integrability/nonintegrability and geometric structures for the 𝐴-rheonomous affine constraints. Our future work on the 𝐴-rheonomous affine constraints are as follows: (i) theoretical analysis and control synthesis of nonholonomic kinematic/dynamic systems subject to 𝐴-rheonomous affine constraints, (ii) applications to the model reduction theory and constrained mechanical system theory, (iii) extensions to more general constraints, for example, fully rheonomous affine constraints of the form: 𝐴(𝑡,𝑞)+𝐵(𝑡,𝑞)𝑞̇=0. In particular, (iii) seems to be quite difficult to solve because not only the rheonomous vector field 𝑋 but also the distribution 𝐷 contain the time variable 𝑡, that is, we have to deal with the time-varying distribution 𝐷(𝑡,𝑞).

Acknowledgment

This work was partially supported by Kurata Grants.