##
**Methods of differential geometry in analytical mechanics.**
*(English)*
Zbl 0687.53001

North-Holland Mathematics Studies, 158. Amsterdam etc.: North-Holland. x, 483 p. $ 110.50; Dfl 210.00 (1989).

The authors of this book study the Lagrangian and Hamiltonian formulations of classical mechanics in terms of modern differential geometric methods. A widely accepted fact is that the proper understanding of classical mechanics requires that the configuration and phase spaces be differentiable manifolds, since nature is in general nonlinear. After reviewing standard geometrical structures of differentiable manifolds, many special structures including almost tangent structures on tangent bundles and symplectic structures on cotangent bundles are analyzed. The former structures are the basis for Lagrangian mechanics, while Hamiltonian mechanics is based on the latter.

A symplectic differential form (which is of order 2 and has maximal rank) is canonically defined on the cotangent bundle \(T^*M\) of a manifold M. Then any real-valued function H on \(T^*M\) produces a unique well- defined vector field \(X_ H\), given locally by Hamilton’s equations for H. This can be defined on any 2-dimensional differentiable manifold possessing a symplectic form. An almost tangent structure on the tangent bundle TM of M is a certain tensor J defined canonically. In contrast with the above situation, not any real valued function L on TM defines a vector field \(X_ L\), unless L is what is called hyperregular Lagrangian (satisfying some convexity conditions along the fiber). The formalism needed to construct the vector field is in fact more complicated than the Hamiltonian case.

Because of the above, and since there are many specialized books treating Hamiltonian structures, the authors emphasize more on the study of Lagrangian structures. This point of view is due to J. Klein [Ann. Inst. Fourier 12, 1-124 (1962; Zbl 0281.49026)] and was adopted in the book by C. Godbillon [Géométrie différentielle et mécanique analytique (Paris 1969; Zbl 0174.246)]. The problem of defining a dynamics in the Lagrangian case when L is not hyperregular is generalized into the concept of Lagrangian submanifold of TTM. This submanifold is locally defined by Euler-Lagrange equations for L on TM, which are not explicit second order differential equations unless L is regular (i.e., locally hyperregular). If L is hyperregular this submanifold is exactly the image of \(X_ L\). The Lagrangian structure for L is connected with a Hamiltonian structure, by means of the associated Legendre transformation Leg: TM\(\to T^*M\), which is not invertible unless L is hyperregular.

The problem when L is not regular is that by using the almost tangent structure we only get on TM a differential 2-form with some degeneracy. This is known as a presymplectic structure. If we wish that Euler- Lagrange equations give a vector field defining the dynamics, we are led into constrained Lagrangians, via restriction to some submanifold of TM. There is a Hamiltonian counterpart to this situation, known as Dirac- Bergmann constraint theory: Given a Hamiltonian function defined on some submanifold of a symplectic manifold, find an extension to the whole manifold such that the corresponding vector field is invariant in the given submanifold.

Related topics of Lagrangian Mechanics are the variational approach, the inverse problem and Noether symmetries. Finally the appendices review particle mechanics in local coordinates, as well as jets and higher order tangent bundles, which are important for mechanical systems where the Lagrangian in local coordinates depends on derivatives of order higher than 1. See the authors’ monograph “Generalized classical mechanics and field theory” [Notas de matematica 102 (North Holland, Amsterdam 1985; Zbl 0581.58015)] for these general Lagrangian systems.

A symplectic differential form (which is of order 2 and has maximal rank) is canonically defined on the cotangent bundle \(T^*M\) of a manifold M. Then any real-valued function H on \(T^*M\) produces a unique well- defined vector field \(X_ H\), given locally by Hamilton’s equations for H. This can be defined on any 2-dimensional differentiable manifold possessing a symplectic form. An almost tangent structure on the tangent bundle TM of M is a certain tensor J defined canonically. In contrast with the above situation, not any real valued function L on TM defines a vector field \(X_ L\), unless L is what is called hyperregular Lagrangian (satisfying some convexity conditions along the fiber). The formalism needed to construct the vector field is in fact more complicated than the Hamiltonian case.

Because of the above, and since there are many specialized books treating Hamiltonian structures, the authors emphasize more on the study of Lagrangian structures. This point of view is due to J. Klein [Ann. Inst. Fourier 12, 1-124 (1962; Zbl 0281.49026)] and was adopted in the book by C. Godbillon [Géométrie différentielle et mécanique analytique (Paris 1969; Zbl 0174.246)]. The problem of defining a dynamics in the Lagrangian case when L is not hyperregular is generalized into the concept of Lagrangian submanifold of TTM. This submanifold is locally defined by Euler-Lagrange equations for L on TM, which are not explicit second order differential equations unless L is regular (i.e., locally hyperregular). If L is hyperregular this submanifold is exactly the image of \(X_ L\). The Lagrangian structure for L is connected with a Hamiltonian structure, by means of the associated Legendre transformation Leg: TM\(\to T^*M\), which is not invertible unless L is hyperregular.

The problem when L is not regular is that by using the almost tangent structure we only get on TM a differential 2-form with some degeneracy. This is known as a presymplectic structure. If we wish that Euler- Lagrange equations give a vector field defining the dynamics, we are led into constrained Lagrangians, via restriction to some submanifold of TM. There is a Hamiltonian counterpart to this situation, known as Dirac- Bergmann constraint theory: Given a Hamiltonian function defined on some submanifold of a symplectic manifold, find an extension to the whole manifold such that the corresponding vector field is invariant in the given submanifold.

Related topics of Lagrangian Mechanics are the variational approach, the inverse problem and Noether symmetries. Finally the appendices review particle mechanics in local coordinates, as well as jets and higher order tangent bundles, which are important for mechanical systems where the Lagrangian in local coordinates depends on derivatives of order higher than 1. See the authors’ monograph “Generalized classical mechanics and field theory” [Notas de matematica 102 (North Holland, Amsterdam 1985; Zbl 0581.58015)] for these general Lagrangian systems.

Reviewer: E.A.Lacomba

### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

58-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis |

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

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

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

70H05 | Hamilton’s equations |

70H03 | Lagrange’s equations |