**” (**

*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**