This book presents modern variational calculus in mechanics and field theories with applications to theoretical physics. It is based on modern mathematical tools, specifically fibred spaces and their jet prolongations, which operate with vector fields and differential forms on foundational structures. The book systematically explains Lagrangian and Hamiltonian mechanics and field theory, with a focused exploration of the underlying structures. Additionally, it addresses the well-known inverse problem of calculus of variations and provides examples illustrating key variational physical theories.

The text is complemented by solved examples from physics and includes exercises designed to help readers master the subject. Aimed at PhD students, postdocs, and interested researchers, this book assumes prior knowledge of mathematical analysis, linear and multilinear algebra, as well as elements of general and theoretical physics for effective engagement with the discussion.

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Jana Musilová is a full Professor of General Physics and Mathematical Physics at the Institute of Theoretical Physics and Astrophysics at the Faculty of Science, Masaryk University. She obtained her PhD (CSc) in Experimental Physics from Masaryk University in 1980. In 1991, she completed her habilitation in Solid State Physics and Acoustics, also at Masaryk University.Her research interests have included slow electron diffraction, modulation techniques for studying the optical properties of semiconductors, and ongoing research in the field of calculus of variations (nonholonomic mechanics), geometric analysis, and mathematical physics.She is also actively engaged in research in the field of physics education, teaching basic and advanced courses in physics and mathematics at Masaryk University.

Pavla Musilová is an Assistant Professor at the Institute of Theoretical Physics and Astrophysics at the Faculty of Science, Masaryk University. She obtained her Master's degrees in Physics, Teaching Physics and Mathematics, and in Mathematics - Algebra and Geometry from Masaryk University in 1999 and 2019, respectively. In 2002, she successfully completed her PhD studies in Theoretical Physics at Masaryk University.Her research is devoted to problems of natural operators on manifolds, pseudoriemannian geometries, and physics education. At the Faculty of Science of Masaryk University, her teaching is focused on courses in mathematical analysis, linear algebra and geometry, multilinear algebra, and differential and integral calculus of forms on manifolds.

Olga Rossi, formerly Krupková (+ 2019), was a full Professor of Mathematics and head of the Department of Mathematics at the Faculty of Sciences of the University of Ostrava. She completed her Master's degree and PhD studies in Theoretical Physics at Charles University in Prague. She worked at the Silesian University in Opava, Palacký University in Olomouc, and, in her later years, at the University of Ostrava. Additionally, she held positions at renowned universities worldwide, including Gent, La Trobe University Melbourne, and Stockholm.Professor Rossi served as the EWM regional coordinator for the Czech Republic and was a member of the EWM-WIM Committee of the European Mathematical Society.Her research focused on global analysis and geometry, geometric mechanics, and geometry and mathematical physics. She continues to be recognized by the global mathematical and physical community as an important expert in her fields of research.