| CODE | MAT3716 | ||||||||
| TITLE | Calculus of Variations and Lagrangian Mechanics | ||||||||
| 今日黑料 LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| EQF/MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 5 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | Calculus of Variations: Classical variational problems: shortest paths, brachistochrone, isoperimetric problems. First variation and Euler–Lagrange equations Free endpoints and boundary conditions Natural boundary conditions and transversality. Constrained variational problems; isoperimetric constraints; Lagrange multipliers. Corner conditions (Weierstrass–Erdmann). Optimal control: Variational problems with control variables and dynamic constraints; Pontryagin-type necessary conditions (Hamiltonian, costate, optimality conditions). Lagrangian mechanics: Action principle; generalized coordinates; Lagrange’s equations. Rigid body motion and precession; rolling constraints; small oscillations and normal modes; impulsive motion. Study-Unit Aims: The aim of this study-unit is to develop a unified understanding of variational principles as a foundation for optimisation in geometry, mechanics, and dynamical systems. Students will learn to formulate and analyse optimisation problems over functions, derive necessary conditions for extrema using the Euler–Lagrange equations, and treat boundary conditions, constraints, and non-smooth solutions in a mathematically coherent way. Particular emphasis is placed on optimal control, where variational ideas underpin modern applications such as trajectory planning in autonomous driving, motion and manipulation in robotics, and dynamic optimisation problems in economics. A concise introduction to Lagrangian mechanics demonstrates how equations of motion arise from action principles and how variational structure provides insight into complex constrained systems. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Formulate geometric, mechanical, and economic problems as variational or optimal control problems with appropriate admissible functions; - Derive and apply the Euler–Lagrange equations, including natural boundary conditions and transversality conditions; - Handle constrained variational problems, including integral constraints and problems with corners, using Lagrange multipliers and Weierstrass–Erdmann conditions; - Formulate basic optimal control problems and interpret Pontryagin-type necessary conditions, including Hamiltonians and costate variables; - Derive Lagrange’s equations of motion from variational principles and identify conserved quantities arising from symmetries; - Analyse representative applications in mechanics, robotics, autonomous systems, and economics using variational and control-theoretic methods. 2. Skills: By the end of the study-unit the student will be able to: - Translate real-world optimisation problems into precise mathematical models with clearly stated assumptions and constraints; - Reason analytically for deriving and interpreting necessary conditions from abstract principles; - Understand trade-offs and work with constrained systems, relevant to decision-making in engineering, economics, and data-driven contexts; - Acquire enhanced problem-solving skills through the synthesis of mathematical theory, physical intuition, and computational or applied perspectives. Main Text/s and any supplementary readings: |
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| STUDY-UNIT TYPE | Lecture and Tutorial | ||||||||
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| LECTURER/S | Joseph Muscat |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2026/7. It may be subject to change in subsequent years. |
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