The geometry of ordinary variational equations
As stated in the preface, the goal of this work is ``to provide a general, comprehensive and self-contained geometric theory of ordinary differential equations which are variational''. The presented intrinsic framework from differential geometry and global analysis allows to handle also nonautonomous and degenerate situations in a unified fashion. The introduction provides motivation for the rest of the text in the form of a review of historical problems and their solutions, starting from the standard derivation of the Euler-Lagrange equations in coordinates, Bernoulli's brachystochrone problem, the inverse problem of the calculus of variations (find the conditions on the coefficients of a second order ordinary differential equation such that the later is variational, i.e. takes the form of an Euler-Lagrange equation) stated by Helmholtz and endowed with a recent proof, the notion of equivalent Lagrangians (i.e. their Euler-Lagrange equations coincide), conditions on a connection so that its geodesics are Euler-Lagrange equations, the form of the potential so that the equations describing a mechanical holonomic system are variational (one sees as a consequence that the origin of the form of the Lorentz force as well as the Coriolis effect is variational), and the derivation of Hamilton's equations in the regular case under a Legendre transformation for integration. Chapter two provides the language and the notations so that the rest of the text can be read. One starts with distributions and their symmetries. Frobenius theorem, and one continues with presymplectic forms, their representations in coordinates via Darboux theorem, and with their symmetries. One defines next the fundamental object, i.e. a jet prolongation of a fibred manifold constructed from its jets of sections. The base space is, in general, one-dimensional and means physically time (this is why the framework is well suited to nonautonomous Lagrangians). In this case coordinate representations are presented. Projectable and vertical vector fields on the jet manifold are defined as well as projectable forms, horizontal forms and the horizontalization mapping with properties, contact forms and the contactization mapping with properties, and finally semisprays. The order of contactness is introduced (it expresses a degree of nonlinearity), and a decomposition of a form into \(i\)-contact forms is provided. Poincaré's Lemma on nice fibred manifolds is shown to hold. Finally, the author defines jet fields, integral sections (their appropriate jet is a jet field), and the conditions they have to satisfy. Chapter three describes Lagrangian dynamics on fibre manifolds. A Lagrangian of order \(r\) on a fibred manifo1d \(\pi:Y\to X\) is a horizontal one-form \(\lambda\) on \(J^rY\), the \(r\)-jet manifold of \(\pi\). The first variation of the defined action functional is obtained by differentiating one-parameter groups generated by a deformation of a section appearing as the quantity sought for in the definition of the action. A Lepagean one-form \(\theta_\lambda\) is introduced. It differs from the Lagrangian by contact forms which are used to annihilate high-order terms of the deformation which appear in the computation of the first variation. These contact forms depend only on the Lagrangian (and can be computed explicitely). \(d \theta_\lambda\) has a decomposition containing a term called the dynamical form \(E_\lambda\), which is also determined by the Lagrangian. The intrinsic Euler-Lagrange equations take the form \[ J^{2r-1}\gamma^*i_\eta d\theta_\lambda= E_\lambda \circ J^{2r} \gamma=0 \] for \(\eta\) on \(J^{2r-1}Y\). Here \(\gamma\) is a section on \(\pi\) which is the sought solution. The author points out that in field theory \((\dim X>0)\) many Lepageans can be associated to the Lagrangian. Chapter four, (``Variational equations'') takes first the point of view of the inverse calculus of variations, i.e. given a dynamical form on \(J^rY\) which is a one-contact element of \(\Lambda^2_Y(J^rY)\), find out if it is an Euler-Lagrange form (an \(E_\lambda)\) in which case it is said to be variational. It is convenient to introduce a Lepagean two-form which is closed with a one-contactization being a dynamical form. It is shown that it is completely determined (locally) by a variational form. So from the characterization in coordinates of Lepagean two-forms, one gets a corresponding one for variational forms which generalize the Helmholtz conditions. One obtains a global variational form if the Lepagean two-form is exact. The author remarks that obstructions to the existence of global Lagrangians are controlled by de Rham cohomology of \(Y\). A canonical representation of a Lepagean two-form (a Lagrangian system) in terms of the computed Hamiltonian and momenta is presented next. This allows to select minimal order Lagrangians from all the equivalent ones of the Lagrangian system in a convenient way. The author remarks that the canonical way of defining the Hamiltonian and the momenta from a Lagrangian of not necessarily minimal order leads to ambiguous quantities. The chapter ends with some formulas which show commutation rules between horizontalization or contactization and isomorphism of fibred manifolds when applied to Lepagean one- and two-forms respectively, and, as a consequence, the author shows the effect of the isomorphism on a Lagrangian (on a variational form, respectively) in their Lepagean equivalents as well as on the Lagrangian in the variational form. Infinitesimal versions of these results (the Lie derivatives) are proved, too. One applies some formulas to the case of holonomic constraints where the isomorphism is the embedding into the manifolds of constraints. The next chapter places beautifully Hamiltonian systems in the previous framework through the following trick. Look at the Lepagean two-form \(\alpha\) on \(J^{s-1}Y\) as a Lagrangian system of order zero on the fibred manifold \(\pi_{s-1}\). The corresponding dynamical form is called the Hamilton form on \(J^1(J^{s-1}Y)\); it is variational with Hamiltonian being the horizontal part (with respect to \(\pi_{s-1})\) of the Lepagean one-form associated to the original system. The local sections \(\delta\) of \(\pi_{s-1}\) are Hamilton extremals given by \(\delta^* i_\eta \alpha=0\) for all (projectable) vector fields \(\eta\) on \(J^{s-1}Y\). This point of view is, however a generalization because the solutions are not necessarily holonomic sections (in other words, they cannot be written as \(\delta =J^{s-1}\gamma\) for an extrema \(\gamma\) of \(E)\). Hamilton extremals are investigated through the characteristic distribution of \(\alpha\). They exist if it is spanned by vector fields with a horizontal component leading to the Euler-Lagrange distribution (and equivalent characterizations are provided). Holonomic Hamilton extremals are characterized in terms of semisprays, too. The next two chapters concentrate on contributions of the author. Chapter six studies regular Lagrangian systems where regularity is defined in a coordinate-free way through the requirement that its Euler-Lagrange distribution has constant rank one. In this case, Hamilton extremals are holonomic one-dimensional trajectories. Several examples show that this extends the usual definition of regularity through the nonsingularity of the Hessian of the Lagrangians, in particular by taking account of various situations in the high-order case. Regularity allows to introduce the Legendre transformation via the momenta, and two cases are distinguished: odd or even order. Coordinate expressions for the Hamilton form, the Hamilton equations and the characteristic distributions are obtained in both cases. Several examples and three points of view concerning the inverse calculus of variations end the chapter. Chapter seven classifies singular Lagrangian systems. A primary constraint set allows to locate those points where the characteristic distribution and the Euler-Lagrange distribution do not coincide, so that some parts of the phase space cannot support a dynamics. In other cases, a semiregular Lagrangian system has an Euler-Lagrange distribution with locally constant rank; the not necessarily holonomic Hamilton extremals are embedded in leaves of a foliation determined by the completely integrable Euler-Lagrange distribution. Next, ``a primary semispray constraint set'' allows to locate those points where no semispray belongs to the Euler-Lagrange distribution excluding the existence of holonomic sections. Weakly regular Lagrangian systems carry no primary semispray constraint. Holonomic sections are integral sections of a subdistribution of the Euler-Lagrange distribution spanned by semisprays. The motion is not deterministic and does not evolve in a leaf unless one assumes semiregularity. The introduced generalized Legendre transformations can be applied to semiregular and first-order weakly regular Lagrangian systems. Chapter eight is devoted to symmetries. One distinguishes point (respectively dynamical) symmetries of a Lagrangian system as symmetries of associated objects (the Lagrangian system itself, the variational form, the Lagrangian, the Lepagean one-form, the characteristic or Euler-Lagrange distribution), where the Lie derivative with respect to a projectable (respectively any) vector field vanishes, and first integrals of the Lagrangian system as first integrals of the characteristic distribution which coincide with the defined constants of motion in the regular case. The various Lie point symmetries are related, and a coordinate-free version of Noether's theorem shows how first integrals arise from Lie point symmetries. Chapter nine (``Geometric integration methods'') introduces a ``complete integral'' of a closed two-form \(\alpha\) on a manifold which is typically a distribution containing the characteristic distribution \({\mathcal D}\) of \(\alpha\) and \(rk{\mathcal D}\) symmetries \(\xi_k\) of \(\alpha\). One obtains a Darboux chart of \(\alpha\) made of first integrals of \({\mathcal D}:\alpha= da^K\wedge db_K\). \(da^K= i_{\xi_k} \alpha\) and the momenta are obtained, too. This is specialized to the case of a fibred manifold \(J^{s-1}Y\) leading to a `` Jacobi complete integral''. More simplifications occur when \(s=2c\) or \(s=2c+1\), where \(\dim Y=c+1\). A function \(S\) which allows to compute the \(b_K\) solves the Hamilton-Jacobi equation. In general, \(S\) is seen as an integrating term which generates invariant transformations of \(\alpha\) (corresponding to the symmetries \(\xi_k)\) transforming Legendre coordinates \((p^i,q_i)\) in Jacobi coordinates \((a^K,b_K)\). Next, for a Lagrangian system \(\alpha_E\) of order \(s-1\), the phase space is \(J^{s-1}Y\) and \(J^{c-1}Y\) is the ``quasi-configuration space''. A section \(\gamma :I\to Y\) is an integral section of a jet field \(\varphi: J^{c-1}Y\to J^{s-1}Y\) if \(\varphi\circ J^{c-1}\gamma =J^{s-1}Y\). The Hamilton-Jacobi equation for fields of extremals is \(\varphi^* \alpha_E=0\). Their integral sections contain the extremals of \(E\), and the converse holds locally for semiregular Lagrangian systems providing equivalence between necessary and sufficient conditions of optimality (the semiregularity hypothesis allows to lift fields on the quasi-configuration space to fields tangent to the leaves of the Jacobi complete integral). A further simplification is obtained by considering \(J^{s-c}Y\) as the quasi-configuration space. The integral sections \(\gamma\) of the projected fields (the so-called Hamilton-Jacobi fields) are such that \(\varphi\circ J^{c-1} \gamma\) is a Hamilton extremal of \(\alpha_E\). We have holonomy up to order \(c-1\) only, and holonomy in the regular case. (Note: it is not clear why precisely \(s-c\) is helpful unless due to technical reasons which may be found in another publication). Next one supposes \(s=2c-1\), so that the two preceding quasi-configuration spaces coincide. The \(\beta_K\) are first integrals of the Hamilton-Jacobi distribution corresponding to the Hamilton-Jacobi fields (!). So, fields of extremals (the first configuration) beautiful coincide with jets of extremals (the second configuration). The Hamilton-Jacobi distribution (integral sections of which are \(J^{c-1}\gamma)\) can be put in correspondence with the Jacobi complete integral (integral sections of which are \(\varphi\circ J^{c-1}\gamma)\). Moreover, one sees that \(S\) foliates the integral manifolds of the Hamilton-Jacobi distribution. The last, chapter examines special cases, in particular metric structures associated to first-order Lagrangian systems. This volume is well written. In the proofs, the details of calculations are sometimes omitted (typical example p. 194, Lemma 9.5.6). On the other hand, many examples illustrate the point, so that the reader develops a better feeling of the subjects. The book can best be appreciated when it is read without skipping chapters (except the last one). Each chapter evolves from concepts presented in the previous ones, forming in this way a harmonious ensemble. The global line of thinking is: language-definitions-classification-structure of solutions via symmetries. There are minor inconsistencies or typographical mistakes. A rich bibliography and a very useful index complete the book. The central role of the Lagrangian system can be seen throughout the book. From it, the relations between the Lagrangian, the Lepagean one-form, the dynamical form are clearly seen. It allows to make clear how the Legendre transformation arises, it allows to study singular situations in a straightforward way, and its symmetries generate first integrals useful for integration. It allows to express fields of extremals. Moreover, the book helps to demonstrate the importance of holonomy and the impact of the choices of configuration spaces. The volume shows how fruitful the interplay between differential geometry and the calculus of variations is. The book is a must for those involved in higher-order Lagrangian dynamics, and it can be recommended to scientists because it provides a language (which can be generalized to field theory and PDEs) for studying variational ODEs, and much more, it allows to realize important and not widespread truths.
inverse problem of calculus of variations
scientific article; zbMATH DE number 1097467