Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus
In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulation...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/18/3231 |
| _version_ | 1797485264401072128 |
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| author | Victor Dods |
| author_facet | Victor Dods |
| author_sort | Victor Dods |
| collection | DOAJ |
| description | In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived. |
| first_indexed | 2024-03-09T23:16:20Z |
| format | Article |
| id | doaj.art-21ba9d8ff82542a0868fca627ddb1eca |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T23:16:20Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-21ba9d8ff82542a0868fca627ddb1eca2023-11-23T17:34:57ZengMDPI AGMathematics2227-73902022-09-011018323110.3390/math10183231Riemannian Calculus of Variations Using Strongly Typed Tensor CalculusVictor Dods0Independent Researcher, Seattle, WA 98106, USAIn this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.https://www.mdpi.com/2227-7390/10/18/3231Riemannian manifoldstensor and tensor field type systemcalculus of variationsEuler–Lagrange equations |
| spellingShingle | Victor Dods Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus Mathematics Riemannian manifolds tensor and tensor field type system calculus of variations Euler–Lagrange equations |
| title | Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus |
| title_full | Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus |
| title_fullStr | Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus |
| title_full_unstemmed | Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus |
| title_short | Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus |
| title_sort | riemannian calculus of variations using strongly typed tensor calculus |
| topic | Riemannian manifolds tensor and tensor field type system calculus of variations Euler–Lagrange equations |
| url | https://www.mdpi.com/2227-7390/10/18/3231 |
| work_keys_str_mv | AT victordods riemanniancalculusofvariationsusingstronglytypedtensorcalculus |