Diffieties and Liouvillian Systems

Liouvillian systems were initially defined in the differential algebra setting. We give here a new formulation using the language of diffieties and infinite dimensional geometries. This mathematical framework is well suited to study Liouvillian systems. Recall that one of the main property of flat systems is that the variables of the system (state, inputs) can be directly expressed, without any integration of differential equations, in terms of the flat output and a finite number of its time derivative. Liouvillian systems share a similar property. To be able to derive the trajectories of a Liouvillian system, we also need some elementary integrations called quadratures. This can be illustrate through the following academic example

It is quite easy to show that (1) is flat for i = 1, 2 and not flat for i = 3 ([2]). However, for i = 3, the subsystem ẋ2 = x 3 , ẋ3 = u.

(

is flat with a flat output y = x 2 and the trajectory of x 1 can be obtained by mean of an elementary integration x 1 = y + ẏ2 .

For the sake of convenience, we first recall, in sections 2 and 3, some facts concerning the theory of diffieties and the Lie-Bäcklund approach to equivalence and flatness (cf [4,5,6,7,13]). In section 4, we define Liouvillian systems using the language of diffieties. Finally, we illustrate the class of Liouvillian through the concrete case of rolling bodies ( [1], [10], [9]).

Let I be a countable set of cardinality ℓ, which may be finite or not, and R I the linear space of all real-valued functions x = (x i ) on I. The space R I has the natural topology of the Euclidean space if I is finite and the Fréchet topology otherwise. The elements x i , i ∈ I, are called coordinates. For an open set U ⊂ R I we denote by C ∞ (U ) the space of all real-valued functions on U that depend on finitely many coordinates and are smooth as functions of a finite number of variables. A chart on a set M is a 3-tuple (U, ϕ, R I ), where U is a subset of M , ϕ is a bijection of U onto an open subset ϕ(U ). The notions of smooth charts and smooth atlases can be defined as in the finite dimensional case. The set M , equipped with an equivalence class of smooth atlases, is called a C ∞ R I -manifold. The number ℓ does not depend on a chart (U, ϕ, R I ) and is called the dimension of the smooth manifold M .

A diffiety is a pair M = (M, CT M ) where M is a C ∞ R I -manifold and CT M a finite dimensional involutive distribution on M . The distribution CT M is called Cartan distribution and its dimension the Cartan dimension of M . Local smooth sections of CT M are called Cartan fields. We are only concerned here with the case of ordinary diffieties, i.e., the dimension of CT M is equal to 1. For the sake of convenience, we use without distinction the notations (M, CT M ) and (M, ∂ M ) to denote the ordinary diffiety M , where ∂ M is a basis vector field of CT M . Let M = (M, CT M ) be a diffiety with dim CT M = 1. Let (U, ϕ, R I ) be a chart on M and ∂ M be a basis vector field of CT M on U , then the 4-tuple (U, ϕ, R I , ∂ M ) is called a chart on M . We denote by ker ϑ M the kernel of the linear map

Let φ : M → N be a smooth mapping. As usual, we denote by φ * : T M → T N the differential (or tangent) mapping of φ, where T M (resp. T N ) is the tangent bundle of M (resp. N ), and by φ * : T * N → T * M the dual differential mapping of φ, i.e., the dual mapping of φ * , where T * M (resp. T * N ) cotangent bundle of M (resp. N ).

A smooth mapping φ :

The diffiety F , as above defined, is usually called trivial diffiety and plays a central role in the Lie-Bäcklund approach of flatness. On some occasions, we will use the short notation

to represent the basis Cartan field ∂ F .

A diffiety M is said to be (locally) of finite type if there exists a (local) Lie-Bäcklund submersion π : M → F such that the fibers are finite dimensional. The integer m is called the (local) differential dimension of M (cf. [5]).

M is a diffiety of finite type where a Cartan field ∂ M has been chosen once for all;

(ii) R is endowed with a canonical structure of a diffiety, with global coordinate t and Cartan field ∂/∂t;

The system σ F = (F , R, pr), where pr is the natural projection mapping pr : {t, w (ν) i } → t and F a trivial diffiety, is called a trivial system. The differential dimension of a system σ M = (M , R, λ), denoted dim diff σ M , is the differential dimension of the associated diffiety M . Definition 3.2 ( [5]). Two systems σ M = (M , R, λ) and σ N = (N , R, δ) are said to be (differentially) equivalent, written σ M ≃ σ N , if and only if

A system σ M = (M , R, λ) is said to be (locally) differentially flat, or simply flat if it is (locally) equivalent to a trivial system. If {t, y

where F 1 j are C ∞ functions on S. Using the short notation, ∂ S can be written under the form

where F 2 j are C ∞ functions on M , F 2 = (F

i | i = 1, . . . , m ; ν ≥ 0} local coordinates on M . Definition 4.2. Let σ M be a differential extension of a flat s

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