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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].

##### MSC:
 58A20 Jets in global analysis 58A12 de Rham theory in global analysis 58J10 Differential complexes 58E30 Variational principles in infinite-dimensional spaces