Noether’s theorems. Applications in mechanics and field theory.

*(English)*Zbl 1357.58002
Atlantis Studies in Variational Geometry 3. Amsterdam: Atlantis Press (ISBN 978-94-6239-170-3/hbk; 978-94-6239-171-0/ebook). xvii, 297 p. (2016).

This book contains a systematic and detailed approach to Lagrangian systems through the variational bicomplex providing a very general geometric and algebraic view of the variational calculus. Noether’s theorems can be seen as the guiding theme for this algebraic approach.

This standpoint is specially apt for the geometric formulation of Lagrangian field theory in which fields are represented by sections of fibre bundles. The calculus of variations of Lagrangians on fibre bundles can be algebraically formulated in terms of the variational bicomplex of differential forms on a jet manifold of sections. In this setting the Lagrangian is a horizontal density.

Chapter 1 contains a description of the space of jets of sections of fibre bundles, the fundamental cohomology properties of the variational bicomplex and the Euler-Lagrange operator. It is also shown that the cohomology of the variational bicomplex yields the variational formula. In chapter 2 it is shown, by using the variational formula, that for any Lagrangian symmetry there is a conserved current whose total differential vanishes on-shell, generalizing Noether’s first theorem from classical mechanics.

Chapter 3 deals with first order Lagrangians and polysymplectic Hamiltonians on Legendre bundles because of their relevance in the physical models. The relationship between both theories is examined under the assumption of regularity conditions.

Chapter 4 is devoted to Lagrangian and Hamiltonian non relativistic mechanics. In this chapter the case of the classical particle motion is seen as the special case in which the base manifold is one dimensional. Particular attention is given to the Kepler problem in chapter 5, providing a detailed analysis of the symmetries characterizing the system.

In chapter 6 Grassmann-graded algebraic calculus, graded manifolds and graded bundles are addressed. The theory of Grassmann-graded Lagrangians on graded bundles is developed in terms of a graded variational bicomplex. This complex provides the appropriate variational formula and gives a very general version of Noether’s first theorem. Chapter 7 contains a detailed analysis of Noether identities in the graded setting. In this context, Noether’s inverse and direct second theorems associate to the Noether identities the gauge symmetries of the Grassmann-graded Lagrangian theory. Chapters 8 and 9 are devoted to applications to Yang-Mills gauge theory on principal bundles and supersymmetric gauge theory on principal graded bundles.

Chapter 10 treats gravitation in the setting of gauge theory on natural bundles and it is shown that in this case the conserved current is the energy-momentum current.

Chapters 11 and 12 deal with topological field theories, namely, Chern-Simons and BF theory.

The book is very well written in a clear and direct style containing very helpful cross references. In addition the book under review includes 4 appendices which make the presentation self contained. Appendix A covers differential calculus over commutative rings, Appendix B includes differential calculus on fibre bundles and appendix C contains a very readable explanation of sheaf cohomology. This material is especially helpful for the most advances chapters. The last appendix covers the Noether identities of differential operators in the homological context.

This standpoint is specially apt for the geometric formulation of Lagrangian field theory in which fields are represented by sections of fibre bundles. The calculus of variations of Lagrangians on fibre bundles can be algebraically formulated in terms of the variational bicomplex of differential forms on a jet manifold of sections. In this setting the Lagrangian is a horizontal density.

Chapter 1 contains a description of the space of jets of sections of fibre bundles, the fundamental cohomology properties of the variational bicomplex and the Euler-Lagrange operator. It is also shown that the cohomology of the variational bicomplex yields the variational formula. In chapter 2 it is shown, by using the variational formula, that for any Lagrangian symmetry there is a conserved current whose total differential vanishes on-shell, generalizing Noether’s first theorem from classical mechanics.

Chapter 3 deals with first order Lagrangians and polysymplectic Hamiltonians on Legendre bundles because of their relevance in the physical models. The relationship between both theories is examined under the assumption of regularity conditions.

Chapter 4 is devoted to Lagrangian and Hamiltonian non relativistic mechanics. In this chapter the case of the classical particle motion is seen as the special case in which the base manifold is one dimensional. Particular attention is given to the Kepler problem in chapter 5, providing a detailed analysis of the symmetries characterizing the system.

In chapter 6 Grassmann-graded algebraic calculus, graded manifolds and graded bundles are addressed. The theory of Grassmann-graded Lagrangians on graded bundles is developed in terms of a graded variational bicomplex. This complex provides the appropriate variational formula and gives a very general version of Noether’s first theorem. Chapter 7 contains a detailed analysis of Noether identities in the graded setting. In this context, Noether’s inverse and direct second theorems associate to the Noether identities the gauge symmetries of the Grassmann-graded Lagrangian theory. Chapters 8 and 9 are devoted to applications to Yang-Mills gauge theory on principal bundles and supersymmetric gauge theory on principal graded bundles.

Chapter 10 treats gravitation in the setting of gauge theory on natural bundles and it is shown that in this case the conserved current is the energy-momentum current.

Chapters 11 and 12 deal with topological field theories, namely, Chern-Simons and BF theory.

The book is very well written in a clear and direct style containing very helpful cross references. In addition the book under review includes 4 appendices which make the presentation self contained. Appendix A covers differential calculus over commutative rings, Appendix B includes differential calculus on fibre bundles and appendix C contains a very readable explanation of sheaf cohomology. This material is especially helpful for the most advances chapters. The last appendix covers the Noether identities of differential operators in the homological context.

Reviewer: Miguel Paternain (Montevideo)

##### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58J90 | Applications of PDEs on manifolds |

58Z05 | Applications of global analysis to the sciences |

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

49Q20 | Variational problems in a geometric measure-theoretic setting |

58A50 | Supermanifolds and graded manifolds |

70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |

58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |