Sugli atti di moto più rigidi possibile

In [3] ([4]) M. Bartolozzi and Aldo Bressan considered a regular portion Δσ of a m-dim. Riemaniann manifold σ=σ(t), possibly moving in a Euclidean space S_v and introduced the notion of (volume preserving) velocity fields as rigid as possible—in short V F R P (V P V F R P)—as an extension of rigid velocity fields —R V F—of classical physics. Existence problems were not considered. In the present paper, written in two parts, we observe that, for V F R P and V P V F R P, and existence theorem similiar to the one for R V F does not hold. Therefore, even if some V F R P (V P V R F P) do always exist, under vevy special constraints, it is natural to look for an alternative definition for the classes V F R P (V P V F R P), so that the existence question has a positive answer in a fully general case. To this end, here in Part 1, a new class V F R P of velocity fields as rigid as possible subject to a set of linear and continuous constraints Γ is introduced. For this class a general existence theorem is proven. A uniqueness result, up to an intrinsecally rigid vector field, is also provided. In part 2 we show how the set of constraints can be expressed in various ways, using integral conditions on volumes or hypersurfaces, essentially equivalent to prescribing the value of the field v_i and of its spin v_i/h at some point P₀. We do this first using a generic field Σ(x)=(e¹ (x), ..., e^(m) (x)) of orthonormal m-tuples of tangent vectors on Δσ. Phisically interesting choices of Σ(x) are then considerd: the ones which are as euclidan (i.e. as parallel) as possible. The existence of such Σ(x) is proven. It is also shown that, in the two-dimensional case, any two fields Σ, Σ’, both as euclidean as possible, differ by a constant rotation.