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Mat. Sb., 1997, Volume 188, Number 1, Pages 59–72 (Mi msb187)  

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.

DOI: https://doi.org/10.4213/sm187

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English version:
Sbornik: Mathematics, 1997, 188:1, 61–74

Bibliographic databases:

UDC: 519.9
MSC: 58E15, 53C07, 49K10
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

Citation in format AMSBIB
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