A Tulczyjew triple for classical fields (Q2881352)

The importance of the Tulczyjew triple in classical mechanics is nowadays widely recognized. The construction consists essentially in a pair of canonical isomorphisms which allow to derive phase equations for mechanical systems and to properly understand the Legendre transformation, even in the case of singular Lagrangians, see, e.g. [\textit{W. M. Tulczyjew}, Symp. math. 14, Geom. simplett., Fis. mat., Teor. geom. Integr. Var. minim., Convegni 1973, 247--258 (1974; Zbl 0304.58015)]. This is a part of the not so well-known Tulczyjew conceptual framework for variational formulations which applies not only to mechanics but to most of classical physics [\textit{W. M. Tulczyjew}, Banach Cent. Publ. 59, 41--75 (2003; Zbl 1074.70010)].NEWLINENEWLINEIn this paper, the author presents a geometric formulation of classical field theory as a remarkable application of the above-mentioned conceptual framework. As a result, she constructs a Tulczyjew triple for first-order classical field theories which allows to formulate intrinsically both Lagrangian and Hamiltonian field equations, with or without external sources. The Legendre transformation is also discussed. Along the way two simple examples are presented: electrostatics and the dynamics of a scalar field.NEWLINENEWLINEIn this approach, one takes the principle of virtual work of statics as a master model for all variational principles of classical physics. The response of a system to a control by external forces, and not just equilibrium configurations is studied. Information on this response is contained in the constitutive set; see the above-cited paper or [\textit{A. De Nicola} and \textit{W. M. Tulczyjew}, Rep. Math. Phys. 58, No. 3, 335--350 (2006; Zbl 1156.70309); \textit{W. M. Tulczyjew}, Commun. Math. 19, No. 2, 179--206 (2011; Zbl 1244.70006)]. The main advantages of Tulczyjew's approach are its generality and flexibility which allow, e.g., to formulate the phase equations in full generality, in mechanics as well as in electrodynamics [\textit{A. De Nicola} and \textit{W. M. Tulczyjew}, Int. J. Geom. Methods Mod. Phys. 6, No. 1, 173--200 (2009; Zbl 1159.49047)], or in first-order general field theory, as in the present paper.NEWLINENEWLINEThe role of the configuration space of statics is played in field theory by the space of fields, which is neither a finite-dimensional manifold nor a Banach one. All the geometric elements needed for the variational calculus must therefore be defined in a suitable way. More treatable objects are recovered when one considers the infinitesimal limit in this framework, e.g. one replaces a finite domain in space-time with an infinitesimal one.NEWLINENEWLINEThe starting point is a locally trivial fibre bundle \(\zeta: E\to M\) on an oriented \(m\)-dimensional manifold \(M\). The fields are represented by the local sections of this bundle, and the bundle of first jets \(J^{1}E\) plays the role of kinematic configurations. A Lagrangian is a map \(L:J^{1}E\to\Lambda^{m}T^{*}M\). The phase space of the theory turns out to be the bundle \(\mathcal{P}=V^{*}E\otimes_{E} \zeta^{*}(\Lambda^{m-1}T^{*}M)\). Here, \(V^{*}E\) is the dual of the vertical subbundle \(VE=\ker T\zeta\) in \(TE\), and \(\zeta^{*}(\Lambda^{m-1}T^{*}M)\) is the pull-back bundle of \(\Lambda^{m-1}T^{*}M\) along the projection \(\zeta\).NEWLINENEWLINEThe Lagrangian side of the Tulczyjew triple is constituted by a morphism of double affine-vector bundles over \(\mathcal{P}\) and \(J^{1}E\): NEWLINE\[NEWLINE \alpha:J^{1}\mathcal{P} \longrightarrow V^{*}J^{1}E\otimes_{J^{1}E} \Lambda^{m}T^{*}M. NEWLINE\]NEWLINE In the absence of external sources, the phase dynamics \(\mathcal{D}\) is the inverse image by \(\alpha\) of the vertical derivative \(dL:J^{1}E\to V^{*}J^{1}E\otimes_{J^{1}E} \Lambda^{m}T^{*}M\) of the Lagrangian, i.e. NEWLINE\[NEWLINE \mathcal D = \alpha^{-1}(dL(J^{1}E))\subset J^{1}\mathcal{P}. NEWLINE\]NEWLINE It is read as an implicit PDE on \(\mathcal{P}\): a (local) section \(\sigma:M\to \mathcal{P}\) is said to be a solution of the Lagrange field equations if the image of its first jet prolongation is contained in \(\mathcal{D}\). Similarly, the Hamiltonian side of the Tulczyjew triple is constituted by a morphism of double affine-vector bundles over \(\mathcal{P}\) and \(J^{1}E\): NEWLINE\[NEWLINE \beta:J^{1}\mathcal{P} \longrightarrow PJ^{\dag}E, NEWLINE\]NEWLINE where \(J^{\dag}E\) is the affine dual of \(J^{1}E\) and \(PJ^{\dag}E\) is the affine phase bundle of the line bundle \(\theta:J^{1}E\to \mathcal{P}\), an affine analog of the cotangent bundle. The phase dynamics \( \mathcal{P}\) can be then alternatively described as the inverse image by \(\beta\) of the affine differential \(dH: \mathcal{P}\to PJ^{\dag}E\) of the Hamiltonian section \(H: \mathcal{P}\to J^{1}E\): NEWLINE\[NEWLINE \mathcal D = \beta^{-1}(dH(\mathcal{P})), NEWLINE\]NEWLINE when such an Hamiltonian exists. In more general cases, one needs a generating family. The spaces \(PJ^{\dag}E\) and \(V^{*}J^{1}E\otimes \Lambda^{m}T^{*}M\) are equipped with canonical 2-forms with values in \(\Lambda^{m}T^{*}M\) that are fiberwise symplectic. On the other hand, the fibers of the bundle \(J^{1}\mathcal{P}\) are endowed with a canonical presymplectic form. The phase space \(\mathcal{P}\) possesses a canonical 1-form with values in \(\Lambda^{m-1}T^{*}M\) which is an analog of the canonical Liouville form on \(T^{*}M\).NEWLINENEWLINEAn alternative formulation of a Tulczyjew triple for classical fields based on (pre-)multisymplectic geometry can be found in [\textit{C. M. Campos} et al., J. Geom. Mech. 4, No. 1, 1--26 (2012; Zbl 1387.70028)].