In recent years, humanity has entered the age of artificial intelligence, and machine learning, powered by massive datasets and computational resources, has been transforming traditional scientific research paradigms. Yet machine learning results often lack interpretability in the scientific sense and fail to meet scientists’ need to uncover objective laws. Greater attention must therefore be paid to whether the scientific problems under investigation contain intrinsically simple structures and to identifying the universal laws underlying them. Mathematically, this means examining the algebraic, geometric, and analytic structures of the models describing scientific problems. These structures not only make research findings scientifically interpretable but also provide a solid foundation for unifying and extending scientific theories. One defining feature of modern physics is its use of modern differential geometry across numerous fields to reveal deep connections underlying physical phenomena. Such connections also exist in mechanics and other engineering sciences grounded in physics.
Modern differential geometry provides a unified description of the nonlinear structures and symmetry relations of complex systems, substantially improving the accuracy of mechanical modeling and the efficiency of control algorithms while revealing multiscale physical laws that conventional methods struggle to identify. Its applications have extended to multibody system dynamics and control, spacecraft dynamics and control, robotic motion control, continuum mechanics, engineering thermophysics, computer graphics, and industrial design, generating fruitful advances at their intersections. Professor Guo Yongxin has long engaged in research and talent development in geometric mechanics, achieving significant results. In response to the urgent need for systematic geometric tools in mechanics and engineering sciences, he offered a two-year series of courses for university faculty and graduate students. Building on these courses, he co-authored Modern Geometric Methods in Mathematical Physics with several collaborators.
Book Summary
Beginning with finite group theory and point-set topology, this book systematically presents the concepts and properties of differentiable manifolds, differential operations on them, and fundamental theorems. Using representative Lie groups and Lie algebras, it details their basic concepts, relationships, adjoint and coadjoint representations, Lie group actions on differentiable manifolds, and classifications of Lie groups and Lie algebras. It also discusses the homotopy and homology groups of differentiable manifolds, integration on manifolds, de Rham cohomology, and the foundations of van Est cohomology. It provides an in-depth analysis of Ehresmann connections on principal bundles (frame bundles) and vector bundles, Cartan's structure equations, holonomy groups of principal bundles, and Koszul connections on tangent bundles, including their curvature and torsion. Finally, it introduces important applications of modern differential geometry in analytical mechanics.
About the Authors
Guo Yongxin holds a Doctor of Engineering degree and is a second-tier professor and doctoral supervisor. He concurrently serves as Director of the Institute of Space Science and Technology at Liaoning University. He receives the State Council Special Government Allowance and has received the First Prize for Natural Sciences of the Science and Technology Awards for Higher Education Institutions, the Second Prize of the National Teaching Achievement Award, and the title of Outstanding Expert of Liaoning Province. He was selected for the Hundred-Talent Level of Liaoning Province's New Century Hundred-Thousand-Ten Thousand Talents Project, named one of Shenyang's Ten Outstanding Young People and an Outstanding Young and Middle-Aged Backbone Teacher of Liaoning Province, and selected for the Outstanding Talents Support Program for Higher Education Institutions in Liaoning Province and the Outstanding Young Scientific Research Talents Training Program of Liaoning Province. He previously served as a member of the Mathematics and Physics Division of the Sixth Science and Technology Committee of the Ministry of Education; Vice Chair of the Dynamics and Control Committee of the Chinese Society of Theoretical and Applied Mechanics; Vice President of the Chinese Society for Interdisciplinary Sciences; a member of the Fifth Discipline Review Group of the Academic Degrees Committee of the Liaoning Provincial People's Government; a Young and Middle-Aged Decision-Making Advisory Expert of Liaoning Province; Vice President of the Liaoning Physical Society; Chair of the Liaoning Provincial University Physics Committee; and Director of the Liaoning Provincial Key Laboratory of Optoelectronic Functional Devices and Detection Technology. He has served on the editorial boards of Communication in Mathematics, Interdisciplinary Science, International Journal of Dynamics and Control, and Journal of Dynamics and Control, as well as the editorial board of the book series Frontiers of Mechanics Fundamentals and Engineering Technology (Higher Education Press).
Professor Guo Yongxin primarily researches geometric methods for dynamical systems and is a leading scholar in analytical mechanics in China. He has published more than 60 representative academic papers, currently leads one Key Program funded by the National Natural Science Foundation of China (NSFC), and has led six completed national-level projects and several completed provincial research projects.
Liu Shixing holds a Doctor of Engineering degree and is a professor, doctoral supervisor, Vice Dean of the School of Physics at Liaoning University, and Deputy Director of the University's Institute of Space Science and Technology. His research focuses on dynamics and control. He has led three General Programs funded by the National Natural Science Foundation of China (NSFC), participated as a subproject leader in one NSFC Key Program, led several completed provincial research projects, and published more than 40 academic papers. He currently serves as a member of the Dynamics and Control Committee of the Chinese Society of Theoretical and Applied Mechanics; Deputy Head of its Analytical Mechanics Group; an external peer reviewer for NSFC projects; an expert in degree and graduate education evaluation for the Ministry of Education; and an editorial board member of Journal of Dynamics and Control.
Translated by: Tan Ruomeng
Edited by: Mark Moore
Proofread by: Mu Xiaolin
Reviewed by: Jin Jinghua
