My new book: Noether’s Theorems
sardanashvily
My new book: “Noether’s Theorems. Applications in Mechanics and Field Theory” (Springer, 2016) has been published (Resume).


In this book, Noether's theorems are presented in the most general and universal form. The calculus of variations and Lagrangian theory are formulated in a very general setting. Relevant examples from mechanics and field theory help the reader to understand the general theory of Noether's theorems and their applications in physic


Contents

1 Calculus of variations on fibre bundles
  1.1 Infinite order jet formalism
  1.2 Variational bicomplex
  1.3 Lagrangian formalism
2 Noether's first theorem
  2.1 Lagrangian symmetries
  2.2 Gauge symmetries: Noether's direct second theorem
  2.3 Noether's first theorem: Conservation laws
3 Lagrangian and Hamiltonian field theories
  3.1 First order Lagrangian formalism
  3.2 Cartan and Hamilton--De Donder equations
  3.3 Noether's first theorem: Energy-momentum currents
  3.4 Conservation laws in the presence of a background field
  3.5 Covariant Hamiltonian formalism
  3.6 Associated Lagrangian and Hamiltonian systems
  3.7 Noether's first theorem: Hamiltonian conservation laws
  3.8 Quadratic Lagrangian and Hamiltonian systems
4 Lagrangian and Hamiltonian nonrelativistic mechanics
  4.1 Geometry of fibre bundles over R
  4.2 Lagrangian mechanics. Integrals of motion
  4.3 Noether's first theorem: Energy conservation laws
  4.4 Gauge symmetries: Noether's second and third theorems
  4.5 Non-autonomous Hamiltonian mechanics
  4.6 Hamiltonian conservation laws: Noether's inverse first theorem
  4.7 Completely integrable Hamiltonian systems
5 Global Kepler problem
6 Calculus of variations on graded bundles
  6.1 Grassmann-graded algebraic calculus
  6.2 Grassmann-graded differential calculus
  6.3 Differential calculus on graded bundles
  6.4 Grassmann-graded variational bicomplex
  6.5 Grassmann-graded Lagrangian theory
  6.6 Noether's first theorem: Supersymmetries
7 Noether's second theorems
  7.1 Noether identities: reducible degenerate Lagrangian systems
  7.2 Noether's inverse second theorem
  7.3 Gauge supersymmetries: Noether's direct second theorem
  7.4 Noether's third theorem: Superpotential
  7.5 Lagrangian BRST theory
8 Yang--Mills gauge theory on principal bundles
  8.1 Geometry of principal bundles
  8.2 Principal gauge symmetries
  8.3 Noether's direct second theorem: Yang-Mills Lagrangian
  8.4 Noether's first theorem: Conservation laws
  8.5 Hamiltonian gauge theory
  8.6 Noether's inverse second theorem: BRST extension
9 SUSY gauge theory on principal graded bundles
10 Gauge gravitation theory on natural bundles
  10.1 Relativity Principle: Natural bundles
  10.2 Equivalence Principle: Lorentz reduced structure
  10.3 Metric-affine gauge gravitation theory
  10.4 Energy-momentum gauge conservation law
  10.5 BRST gravitation theory
11 Chern-Simons topological field theory
12 Topological BF theory
A Differential calculus over commutative rings
  A.1 Commutative algebra
  A.2 Differential operators on modules and rings
  A.3 Chevalley--Eilenberg differential calculus
  A.4 Differential calculus over C^\infty (X). Serre-Swan theorem
B Differential calculus on fibre bundles
  B.1 Geometry of fibre bundles
  B.2 Jet manifolds
  B.3 Connections on fibre bundles
  B.4 Higher order jet manifolds
  B.5 Differential operators and equations
C Calculus on sheaves
  C.1 Sheaf cohomology
  C.2 Abstract de Rham theorem
  C.3 Local-ringed spaces
D Noether identities of differential operators

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