Mathematical Foundations and Numerical Analysis of the Dynamics of an Isotropic Universe

本書是《Fondamenti matematici e analisi numerica della dinamica di un Universo isotropo》一文的增強版和擴展版英文版，該文由都靈科學院於2018-2019年出版，編號為42-43卷。本書總結了一個長期宇宙學研究項目的一些主要發現，旨在通過清晰的數學公設來闡明重要結果。儘管付出了努力，但由於早期宇宙的複雜性和眾多觀測參數，準確描述宇宙演化的單一數學模型仍然難以捉摸。在過去的世紀中，提出並淘汰了各種模型，這在宇宙學常數和空間曲率假設的辯論中得到了體現。目前，許多模型缺乏明確的基礎，導致該領域的混亂。標準的宇宙學方法依賴於如Weyl原則、均勻性和各向同性等原則，但可能忽略了純幾何性質與依賴於場方程的性質之間的區別。本書旨在通過從可理解的數學公設出發，帶來宇宙學的秩序，並引導至定理。對公設的分歧可以促使調整或替代方法。物理學通常由缺乏明確界定的基本概念和公設的演繹理論組成，這一批評對宇宙學理論同樣適用。儘管在1990年代末對Lambda冷暗物質模型達成了一致，但文獻中缺乏邏輯演繹結構使理解變得複雜，導致一些人幽默地稱之為「擴張的宇宙與擴張的混亂」。

Sergio Benenti graduated in mathematics with honors in 1968. A year later he became a permanent assistant professor of Rational Mechanics, and in 1980 he was appointed full professor of Rational Mechanics. In 2021 he was awarded the title of professor emeritus. He has been a member of the Turin Academy of Sciences since 1987. His scientific activity has covered several areas of research. First, the separation of the variables of the Hamilton-Jacobi equation, solving by geometrical methods the long-standing problem posed by Levi-Civita. These results, known as Benenti systems, Benenti theorem, and Benenti tensor, have been widely accepted by researchers and have led to numerous publications. He also applied his theory to exact solutions of Einstein's equations, providing particular results that later proved to be useful in the study of black holes. Meanwhile, he worked extensively on symplectic geometry and its applications to mathematical physics. The Steklof Math Institute published (in Russian, 2009) the text of a course held in Turin on this topic. This book was followed by a revisited and expanded English version, published by Springer under the title Hamiltonian Structures and Generating Families (2011). He recently explored nonholonomic mechanical systems, providing a correct and general setting of Gauss's Principle. In addition, he presented the first feasible example of a nonlinear nonholonomic system, showing the inconsistency of several examples so far present in literature. Finally, in the last years, he focused on mathematical models of cosmology, which is currently his main area of research.

Sergio Benenti 於1968年以優異成績畢業於數學系。一年後，他成為理性力學的專任助理教授，並於1980年被任命為理性力學的正教授。2021年，他獲得名譽教授的稱號。自1987年以來，他一直是都靈科學院的成員。他的科學活動涵蓋了多個研究領域。首先，他研究了哈密頓-雅可比方程的變數分離，並通過幾何方法解決了由Levi-Civita提出的長期問題。這些結果被稱為Benenti系統、Benenti定理和Benenti張量，已被研究人員廣泛接受，並導致了許多出版物。他還將他的理論應用於愛因斯坦方程的精確解，提供了特定的結果，這些結果後來被證明在黑洞研究中非常有用。與此同時，他在辛幾何及其在數學物理中的應用方面進行了廣泛的研究。Steklof數學研究所於2009年（以俄文出版）發表了在都靈舉辦的相關課程的文本。這本書隨後推出了經過修訂和擴展的英文版，由Springer出版，書名為《Hamiltonian Structures and Generating Families》（2011年）。最近，他探索了非完整機械系統，提供了高斯原理的正確和一般設定。此外，他展示了第一個可行的非線性非完整系統的例子，顯示了文獻中幾個例子的矛盾。最後，在最近幾年中，他專注於宇宙學的數學模型，這目前是他的主要研究領域。