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On the connection generated by the problem of minimizing a multiple integral
M. I. Zelikin M. V. Lomonosov Moscow State University
Abstract:
We consider the Dirichlet integral, a functional which depends quadratically on the sections of a vector bundle $\pi \colon \xi \to \mathfrak N$ over a smooth manifold $\mathfrak N$. We obtain and investigate a quadratic system of partial differential equations, called the Riccati equation by analogy with the one-dimensional case. We show that the solutions of this system define a connection $\nabla$ on the bundle $\xi$. A field of extremals for the Dirichlet functional exists if and only if there is a solution of the Riccati equation that defines a flat connection. The existence of a globally defined solution of the Riccati equation satisfying certain additional conditions guarantees that the Dirichlet functional is positive-definite.
Received: 16.04.1996
Citation:
M. I. Zelikin, “On the connection generated by the problem of minimizing a multiple integral”, Mat. Sb., 188:1 (1997), 59–72; Sb. Math., 188:1 (1997), 61–74
Linking options:
https://www.mathnet.ru/eng/sm187https://doi.org/10.1070/sm1997v188n01ABEH000187 https://www.mathnet.ru/eng/sm/v188/i1/p59
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Abstract page: | 434 | Russian version PDF: | 181 | English version PDF: | 24 | References: | 92 | First page: | 3 |
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