## Abstract

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 P_{0}. We do this first using a generic field Σ(x)=(e^{1} (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.

Original language | Italian |
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Pages (from-to) | 60-68 |

Number of pages | 9 |

Journal | Rendiconti del Circolo Matematico di Palermo |

Volume | 32 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1983 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)