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On the normal forms for Pfaffian systems. (English) Zbl 1128.58002
Let $$M$$ be a $$C^\infty$$-manifold of dimension $$n$$ and $$T^*M$$ its cotangent bundle. Let $$C^\infty(M)$$ denote the ring of $$C^\infty$$-functions on $$M$$ and $$\Gamma(T^*M)$$ the $$C^\infty(M)$$-module of global $$C^\infty$$-sections of $$T^*M$$. A $$C^\infty(M)$$-submodule of $$\Gamma(T^*M)$$ is called a Pfaffian system $$S$$ on $$M$$. Let $$J^r(\mathbb R^h,\mathbb R^q)=(x_{\alpha_1},z^i,z^1_{\alpha_1},\dots,z^1_{\alpha_1\dots\alpha_r})$$ be the jet manifold.
There is a canonical Pfaffian system on $$J^r(\mathbb R^h,\mathbb R^q)$$, called the contact system, which is the Pfaffian system generated by the 1-forms $$\omega^i = dz^i - \sum_{\alpha_1=1}^hz^i_{\alpha_1}dx_{\alpha_1}$$, $$\omega^i_{\alpha_1} = dz^i_{\alpha_1} - \sum_{\alpha_2=1}^hz^i_{\alpha_1\alpha_2}dx_{\alpha_2}$$, $$\dots$$, $$\omega^i_{\alpha_1\dots\alpha_{r-1}} =dz^i_{\alpha_1\dots\alpha_{r-1}} -\sum_{\alpha_r=1}^hz^1_{\alpha_1\dots\alpha_r}dx_{\alpha_r}$$. A contact system restricted to a submanifold of $$J^r(\mathbb R^h,\mathbb R^q)$$ is called the Pfaffian system associated to a system of partial differential equations.
In this paper, the author studies local normal forms of Pfaffian systems and gives a necessary and sufficient condition for a Pfaffian system to be transformed into the contact system on the jet manifold $$J^r(\mathbb R^h,\mathbb R^q)$$ or into the Pfaffian system associated to a system of partial differential equations. The properties of relative polarization are used to prove this result which is the generalization of the Darboux theorem on a Pfaffian equation of constant class.
##### MSC:
 58A17 Pfaffian systems 37J55 Contact systems 58A20 Jets in global analysis 70H07 Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics
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