Calculus of variations 2

This book explores classical variational calculus, appealing to analysts, geometers, and physicists. Volume 1 focuses on the formal apparatus of variational calculus and nonparametric field theory, while Volume 2 addresses parametric variational problems, Hamilton-Jacobi theory, and classical first-order partial differential equations. Future discussions will cover developments from Hilbert's 19th and 20th problems, emphasizing direct methods and regularity theory. The text highlights the often-overlooked theory of inner variations—variations of independent variables—which provides valuable insights such as monotonicity formulas, conformality relations, and conservation laws. The combined variation of dependent and independent variables leads to Emmy Noether's general conservation laws, a critical tool for exploiting symmetries. Additional sections delve into Legendre-Jacobi theory and field theories, offering a thorough examination of one-dimensional field theory for both nonparametric and parametric integrals, and its connections to Hamilton-Jacobi theory, geometrical optics, and point mechanics. The book also investigates various approaches to utilizing convexity in the calculus of variations, with field theory presenting a particularly nuanced method. Furthermore, it emphasizes the significance of the concept of a null Lagrangian, which plays a crucial role in several contexts.