Generalization of Calculus of Variations for cases where the Lagrangian is non-differentiable
(2023) In Bachelor’s Theses in Mathematical Sciences MATK11 20231Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- Calculus of variations is a branch of mathematics that was developed in the 18th century to solve problems of determining optimal trajectories, shapes and structures subject to constraints. It deals with the minimization of a certain type of integral functional. In the early days of the theory, the smoothness of the functions involved was assumed but, during the following centuries, advances in analysis allowed for more general cases to be considered. The second half of the 20th century saw significant progress in the field, largely driven by the development of optimal control theory, which is widely used in engineering to regulate machines and large systems. New applications in diverse fields such as engineering, economics, mechanics and... (More)
- Calculus of variations is a branch of mathematics that was developed in the 18th century to solve problems of determining optimal trajectories, shapes and structures subject to constraints. It deals with the minimization of a certain type of integral functional. In the early days of the theory, the smoothness of the functions involved was assumed but, during the following centuries, advances in analysis allowed for more general cases to be considered. The second half of the 20th century saw significant progress in the field, largely driven by the development of optimal control theory, which is widely used in engineering to regulate machines and large systems. New applications in diverse fields such as engineering, economics, mechanics and physics, medicine, renewable resources and branches of analysis such as geometry and differential equations, necessitated generalizations in the theory of calculus of variations for dealing with more complex integrands. In this thesis we look at modern generalizations of the theory of calculus of variations for cases where the integrand (Lagrangian) is non-differentiable. In particular, we examine the proof for deriving the modern version of Euler-Lagrange equation, which provides the necessary condition that a minimizing function in the context of calculus of variations should satisfy, and is the cornerstone of the theory. (Less)
- Popular Abstract (Swedish)
- I syfte att förstå lösningar för nutida optimeringsproblem, ska vi se på det matematiska ramverket för den moderna teorin om variationskalkyl. Variationskalkyl är en gren av matematiken som utvecklades under 1700-talet för att lösa problem gällande fastställning av optimala trajektoria, former och strukturer, givet vissa begränsningar.
Dess utveckling pågick parallellt med framsteg inom matematisk analys och betydande genombrott gjordes under andra halvan av 1900-talet, till stor del på grund av utvecklingen av styrteori, som är välanvänt inom teknikvetenskap för maskinreglering och för att optimera stora system.
Den moderna teorin, introducerad av den Kanadensiske matematikern Francis Clarke, för att lösa problem som innefattar mer... (More) - I syfte att förstå lösningar för nutida optimeringsproblem, ska vi se på det matematiska ramverket för den moderna teorin om variationskalkyl. Variationskalkyl är en gren av matematiken som utvecklades under 1700-talet för att lösa problem gällande fastställning av optimala trajektoria, former och strukturer, givet vissa begränsningar.
Dess utveckling pågick parallellt med framsteg inom matematisk analys och betydande genombrott gjordes under andra halvan av 1900-talet, till stor del på grund av utvecklingen av styrteori, som är välanvänt inom teknikvetenskap för maskinreglering och för att optimera stora system.
Den moderna teorin, introducerad av den Kanadensiske matematikern Francis Clarke, för att lösa problem som innefattar mer komplicerade funktioner, har vitt skilda användningsområden som inkluderar teknikvetenskap, ekonomi, mekanik och fysik, medicin och förnybara resurser, samt flertalet grenar inom matematisk analys.
Vi undersöker beviset för den moderna versionen av Euler-Lagranges ekvationen, vilket ger villkoret som en minimeringsfunktion ska uppfylla i kontexten för variationskalkyl och är en hörnsten av teorin. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9133275
- author
- Efstathiou, Zoi LU
- supervisor
- organization
- alternative title
- Generalisering av variationskalkyl till fall där Lagrangianen inte är differentierbar
- course
- MATK11 20231
- year
- 2023
- type
- M2 - Bachelor Degree
- subject
- keywords
- Calculus of variations, optimization, Euler-Lagrange equation, Euler inclusion, Weierstrass necessary condition, nonsmooth analysis, Lagrangian.
- publication/series
- Bachelor’s Theses in Mathematical Sciences
- report number
- LUNFMA-4149-2023
- ISSN
- 1654-6229
- other publication id
- 2023:K22
- language
- English
- id
- 9133275
- date added to LUP
- 2025-06-27 15:53:23
- date last changed
- 2025-06-27 15:53:23
@misc{9133275,
abstract = {{Calculus of variations is a branch of mathematics that was developed in the 18th century to solve problems of determining optimal trajectories, shapes and structures subject to constraints. It deals with the minimization of a certain type of integral functional. In the early days of the theory, the smoothness of the functions involved was assumed but, during the following centuries, advances in analysis allowed for more general cases to be considered. The second half of the 20th century saw significant progress in the field, largely driven by the development of optimal control theory, which is widely used in engineering to regulate machines and large systems. New applications in diverse fields such as engineering, economics, mechanics and physics, medicine, renewable resources and branches of analysis such as geometry and differential equations, necessitated generalizations in the theory of calculus of variations for dealing with more complex integrands. In this thesis we look at modern generalizations of the theory of calculus of variations for cases where the integrand (Lagrangian) is non-differentiable. In particular, we examine the proof for deriving the modern version of Euler-Lagrange equation, which provides the necessary condition that a minimizing function in the context of calculus of variations should satisfy, and is the cornerstone of the theory.}},
author = {{Efstathiou, Zoi}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor’s Theses in Mathematical Sciences}},
title = {{Generalization of Calculus of Variations for cases where the Lagrangian is non-differentiable}},
year = {{2023}},
}