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**The geometry of Hamilton and Lagrange spaces.**
*(English)*
Zbl 1001.53053

Fundamental Theories of Physics. 118. Dordrecht: Kluwer Academic Publishers. xv, 338 p. (2001).

The book is divided into two parts: Hamilton and Lagrange spaces (Chapters 1-8) and Hamilton space of higher order (Chapters 9-13, written by R. Miron). The notion of Lagrange space was introduced in 1974 by J. Kern as a generalization of Finsler space. A Lagrange space is a pair \(L^n= (M,L(x,y))\) formed by a smooth real \(n\)-dimensional manifold \(M\) and a regular Lagrangian \(L(x,y)\). The latter is a mapping \(L:(x,y)\in TM\to L(x,y)\in\mathbb{R}\) of class \(C^\infty\) on \(TM\setminus \{0\}\), where \(\{0\}\) means the null section of \(\pi: TM\to M\), and continuous on \(\{0\}\). Moreover, it is supposed that the tensor field \(g_{ij}= {1\over 2}{\partial L(x,y)\over \partial y^i\partial y}\) on \(TM \setminus\{0\}\) is regular and of constant signature. This gives an excellent geometric model for some important problems in relativity, gauge theory, and electromagnetism. Any Finsler space \(F^n=(M,F(x,y))\) is a Lagrange space \(L^n= (M,F^2)\).

The dual notion is the Hamilton space \(H^n=(M,H(x,p))\), introduced in 1987 by R. Miron; here \(H:T^*M\to\mathbb{R}\) is a regular Hamilton function, satisfying analogous conditions. The case when \(H\) is the square of a function on \(T^*M\), positively 1-homogeneous with respect to the momentum \(p_i\), provides an important subclass of Cartan spaces \(C^n=(M,K (x,p))\), which are dual of the Finsler spaces. This gives a geometric framework for the Hamiltonian theory of mechanics or physical fields.

During the last two decades many mathematical models from Lagrangian mechanics, theoretical physics and variational calculus systematically used multivariate Lagrangians of higher-order acceleration. This led to the geometry of higher order Lagrangian spaces, summarized in the monograph by the first author: The Geometry of Higher-Order Lagrange Spaces. Kluwer Fundamental Theories of Physics 82 (1997; Zbl 0877.53001). In the second part of the present book the same is made for the generalized Hamilton spaces of order 2, in particular for the Cartan spaces of order 2.

The dual notion is the Hamilton space \(H^n=(M,H(x,p))\), introduced in 1987 by R. Miron; here \(H:T^*M\to\mathbb{R}\) is a regular Hamilton function, satisfying analogous conditions. The case when \(H\) is the square of a function on \(T^*M\), positively 1-homogeneous with respect to the momentum \(p_i\), provides an important subclass of Cartan spaces \(C^n=(M,K (x,p))\), which are dual of the Finsler spaces. This gives a geometric framework for the Hamiltonian theory of mechanics or physical fields.

During the last two decades many mathematical models from Lagrangian mechanics, theoretical physics and variational calculus systematically used multivariate Lagrangians of higher-order acceleration. This led to the geometry of higher order Lagrangian spaces, summarized in the monograph by the first author: The Geometry of Higher-Order Lagrange Spaces. Kluwer Fundamental Theories of Physics 82 (1997; Zbl 0877.53001). In the second part of the present book the same is made for the generalized Hamilton spaces of order 2, in particular for the Cartan spaces of order 2.

Reviewer: Ülo Lumiste (Tartu)

### MSC:

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

53D99 | Symplectic geometry, contact geometry |

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

53C80 | Applications of global differential geometry to the sciences |

70H50 | Higher-order theories for problems in Hamiltonian and Lagrangian mechanics |