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**The geometry of higher-order Lagrange spaces. Applications to mechanics and physics.**
*(English)*
Zbl 0877.53001

Fundamental Theories of Physics. 82. Dordrecht: Kluwer Academic Publishers. xv, 332 p. (1997).

This book is devoted to the study of the geometry of higher order osculator bundles, in the author’s terminology, or tangent bundles of higher order in the usual terminology of higher order mechanics. As is well-known, they are the geometric setting for higher order Lagrangian mechanics. In fact, if \(L\) is a Lagrangian function depending on the derivatives up to order \(k\), then it can be considered as a function defined on the tangent bundle of the same order of the configuration manifold \(Q\), that is, \(L:T^kQ \to\mathbb{R}\). So, the dynamics takes place on the tangent bundle of order \(2k-1\) in order to provide Euler-Lagrange equations of order \(2k\), according to the Ostrogradski formalism which was put in symplectic terms in [Refs. 50 and 164]. A pair \((Q,L)\) is a Lagrange space in the author’s terminology. It must be noticed that the author considers Lagrangian functions such that its Hessian matrices have signature. The reader familiar with mechanics should bear this fact in mind.

The author studies the “canonical geometry” provided by the higher order almost tangent structure. Special attention is devoted to discuss higher order connections and the polynomial structures related with them. The prolongation of Riemannian and Finslerian structures is also widely studied as well as the prolongation of subspaces, and the corresponding generalizations of the Gauss-Weingarten and Gauss-Codazzi equations are obtained.

The author also discusses symmetries and Noether theorems. However, recent results are not included (see for instance D. Martín de Diego and M. de León, J. Math. Phys. 36, 4138-4161 (1995; Zbl 0845.70012)] and the references therein, with results by G. Prince, M. Crampin, J. F. Cariñena, E. Martínez, W. Sarlet, F. Cantrijn, G. Marmo and others).

The book is divided into twelve chapters and treats separately the cases of first order, second order and higher order. However, the book is not repetitive.

The author studies the “canonical geometry” provided by the higher order almost tangent structure. Special attention is devoted to discuss higher order connections and the polynomial structures related with them. The prolongation of Riemannian and Finslerian structures is also widely studied as well as the prolongation of subspaces, and the corresponding generalizations of the Gauss-Weingarten and Gauss-Codazzi equations are obtained.

The author also discusses symmetries and Noether theorems. However, recent results are not included (see for instance D. Martín de Diego and M. de León, J. Math. Phys. 36, 4138-4161 (1995; Zbl 0845.70012)] and the references therein, with results by G. Prince, M. Crampin, J. F. Cariñena, E. Martínez, W. Sarlet, F. Cantrijn, G. Marmo and others).

The book is divided into twelve chapters and treats separately the cases of first order, second order and higher order. However, the book is not repetitive.

Reviewer: M.de León (Madrid)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |

70Hxx | Hamiltonian and Lagrangian mechanics |