On null Lagrangians |
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Authors: | M Crampin DJ Saunders |
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Institution: | Department of Applied Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK |
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Abstract: | We consider multiple-integral variational problems where the Lagrangian function, defined on a frame bundle, is homogeneous. We construct, on the corresponding sphere bundle, a canonical Lagrangian form with the property that it is closed exactly when the Lagrangian is null. We also provide a straightforward characterization of null Lagrangians as sums of determinants of total derivatives. We describe the correspondence between Lagrangians on frame bundles and those on jet bundles: under this correspondence, the canonical Lagrangian form becomes the fundamental Lepage equivalent. We also use this correspondence to show that, for a single-determinant null Lagrangian, the fundamental Lepage equivalent and the Carathéodory form are identical. |
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Keywords: | 53C60 70S05 |
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