On a class of polynomial Lagrangians.

*(English)*Zbl 1002.58003
Slovák, Jan (ed.) et al., The proceedings of the 20th winter school “Geometry and physics”, Srní, Czech Republic, January 15-22, 2000. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 66, 147-159 (2001).

This paper is devoted to investigate the variational properties of special Lagrangians and related geometric objects (Euler-Lagrange and Poincaré-Cartan forms, momenta, etc.). The basic tool in this study is the theory of variational sequences of finite order, following D. Krupka [in: Proceedings of Differential Geometry and its Applications (Brno, 1989), 236-254, World Scientific, Singapore (1990; Zbl 0813.58014)].

Let \(\pi:Y\longrightarrow X\) be a fibration, where \(\dim X=n\), and denote by \(J_rY\) the space of \(r\)-jets of local sections of \(\pi\). The \(r\)-th order lagrangian densities which are obtained as the horintalization of \(n\)-forms on \(J_{r-1}Y\) are called special Lagrangians; the authors show that they have polynomial coefficients in the highest order derivatives.

Let us introduce some notation: \(\Lambda_r^k\) will be the sheaf of differential \(k\)-forms on \(J_rY\), \({\mathcal H}_r^k\) the sheaf of horizontal forms, \({\mathcal H}_r^{k,h}\) the sheaf of horizontalized \(k\)-forms belonging to \(\Lambda_{r-1}^k\) (the image of \(\Lambda_{r-1}^k\) by the horizontalization operator \(h\)) and \({\mathcal C}_r^1\) the sheaf of contact \(1\)-forms in \(J_rY\). A form \(\gamma=h(\alpha)\in{\mathcal C}_r^1\wedge{\mathcal H}_{r-1}^{n,h}\) is called a generating form; the decomposition formula given by I. Kolář [J. Geom. Phys. 1, No. 2, 127-137 (1984; Zbl 0595.58016)] shows that each generating form \(\gamma\) can be written as \(\gamma=E_\gamma+d_Hp_\gamma\), where \(d_H\) is the horizontal differential, \(E_\gamma\) contains \(0\)-th order contact forms and \(p_\gamma\) is a \(1\)-contact form called the momentum of \(\gamma\).

Let \(\lambda\) be an \(r\)-th order Lagrangian; if we apply the decomposition formula to \(d\lambda\) we obtain \(d\lambda=E_{d\lambda}+d_Hp_{d\lambda}\), where \(E_{d\lambda}\) and \(p_{d\lambda}\) are the Euler-Lagrange and the momentum form of \(\lambda\), respectively. If \(\lambda=h(\beta)\) is a special Lagrangian, then \(E_{d\lambda}=E_{h(d\beta)}\); furthermore, \(E_{d\lambda}\) is defined in \(J_{2r-1}Y\) and not only in \(J_{2r}Y\), as usual.

Next, the authors describe some general properties of the momenta of generating forms \(h(\alpha)\in{\mathcal C}_r^1\wedge{\mathcal H}_{r+1}^{n,h}\) and their relationship with momenta of special Lagrangians. In particular, it is shown that \(p_{h(\alpha)}\) is unique when \(n=1\) or \(\alpha\in\Lambda_1^n\), and it can be chosen in a natural way when \(\alpha\in\Lambda_2^n\).

Let \(\lambda=h(\beta)\in{\mathcal H}_{r+1}^{n,h}\) be a special Lagrangian; the authors prove that \(h(d\beta)=h(d_Hv(\beta))+d\lambda\), where \(v(\beta)\) is the vertical part of \(\beta\). As a consequence \(p_{d\lambda}\) and \(p_{h(d\beta)}\) can be chosen to be equal if \(h(d_Hv(\beta))=0\); since each \(r\)-th order general Lagrangian \(\lambda\) is an \((r+1)\)-th order special Lagrangian, because \(\lambda=h(\lambda)\) when it is considered as belonging to \(\Lambda_{r+1}^n\), the momenta \(p_{h(d\lambda)}\) and \(p_{d\lambda}\) can be chosen to be equal.

If \(\lambda\) is an \(r\)-th order Lagrangian, a Poincaré-Cartan form is defined as \(\theta=\lambda+p_{d\lambda}\in\Lambda_{2r-1}^n\); for a special Lagrangian a unique class of Poincaré-Cartan forms can be chosen if \(\theta\) is required to fulfill some concitions: \(h(\theta)=\lambda,v(\theta)\in {\mathcal C}_{2r}^1\wedge{\mathcal H}_{2r}^{n-1}\) and \(h(d\theta)=E_{h(d\theta)}\).

Finally, the general results are illustrated with an important and simple example of a special Lagrangian, namely, the Einstein-Hilbert Lagrangian.

For the entire collection see [Zbl 0961.00020].

Let \(\pi:Y\longrightarrow X\) be a fibration, where \(\dim X=n\), and denote by \(J_rY\) the space of \(r\)-jets of local sections of \(\pi\). The \(r\)-th order lagrangian densities which are obtained as the horintalization of \(n\)-forms on \(J_{r-1}Y\) are called special Lagrangians; the authors show that they have polynomial coefficients in the highest order derivatives.

Let us introduce some notation: \(\Lambda_r^k\) will be the sheaf of differential \(k\)-forms on \(J_rY\), \({\mathcal H}_r^k\) the sheaf of horizontal forms, \({\mathcal H}_r^{k,h}\) the sheaf of horizontalized \(k\)-forms belonging to \(\Lambda_{r-1}^k\) (the image of \(\Lambda_{r-1}^k\) by the horizontalization operator \(h\)) and \({\mathcal C}_r^1\) the sheaf of contact \(1\)-forms in \(J_rY\). A form \(\gamma=h(\alpha)\in{\mathcal C}_r^1\wedge{\mathcal H}_{r-1}^{n,h}\) is called a generating form; the decomposition formula given by I. Kolář [J. Geom. Phys. 1, No. 2, 127-137 (1984; Zbl 0595.58016)] shows that each generating form \(\gamma\) can be written as \(\gamma=E_\gamma+d_Hp_\gamma\), where \(d_H\) is the horizontal differential, \(E_\gamma\) contains \(0\)-th order contact forms and \(p_\gamma\) is a \(1\)-contact form called the momentum of \(\gamma\).

Let \(\lambda\) be an \(r\)-th order Lagrangian; if we apply the decomposition formula to \(d\lambda\) we obtain \(d\lambda=E_{d\lambda}+d_Hp_{d\lambda}\), where \(E_{d\lambda}\) and \(p_{d\lambda}\) are the Euler-Lagrange and the momentum form of \(\lambda\), respectively. If \(\lambda=h(\beta)\) is a special Lagrangian, then \(E_{d\lambda}=E_{h(d\beta)}\); furthermore, \(E_{d\lambda}\) is defined in \(J_{2r-1}Y\) and not only in \(J_{2r}Y\), as usual.

Next, the authors describe some general properties of the momenta of generating forms \(h(\alpha)\in{\mathcal C}_r^1\wedge{\mathcal H}_{r+1}^{n,h}\) and their relationship with momenta of special Lagrangians. In particular, it is shown that \(p_{h(\alpha)}\) is unique when \(n=1\) or \(\alpha\in\Lambda_1^n\), and it can be chosen in a natural way when \(\alpha\in\Lambda_2^n\).

Let \(\lambda=h(\beta)\in{\mathcal H}_{r+1}^{n,h}\) be a special Lagrangian; the authors prove that \(h(d\beta)=h(d_Hv(\beta))+d\lambda\), where \(v(\beta)\) is the vertical part of \(\beta\). As a consequence \(p_{d\lambda}\) and \(p_{h(d\beta)}\) can be chosen to be equal if \(h(d_Hv(\beta))=0\); since each \(r\)-th order general Lagrangian \(\lambda\) is an \((r+1)\)-th order special Lagrangian, because \(\lambda=h(\lambda)\) when it is considered as belonging to \(\Lambda_{r+1}^n\), the momenta \(p_{h(d\lambda)}\) and \(p_{d\lambda}\) can be chosen to be equal.

If \(\lambda\) is an \(r\)-th order Lagrangian, a Poincaré-Cartan form is defined as \(\theta=\lambda+p_{d\lambda}\in\Lambda_{2r-1}^n\); for a special Lagrangian a unique class of Poincaré-Cartan forms can be chosen if \(\theta\) is required to fulfill some concitions: \(h(\theta)=\lambda,v(\theta)\in {\mathcal C}_{2r}^1\wedge{\mathcal H}_{2r}^{n-1}\) and \(h(d\theta)=E_{h(d\theta)}\).

Finally, the general results are illustrated with an important and simple example of a special Lagrangian, namely, the Einstein-Hilbert Lagrangian.

For the entire collection see [Zbl 0961.00020].

Reviewer: J.Rodríguez (Salamanca)