| Code |
14806
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| Year |
3
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| Semester |
S2
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Real Analysis and Differential Equations (Calculus I, II and III)
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Learning outcomes |
On the completion of this course, students should be able of:
1. give an account of the foundations of calculus of variations and of its applications in mathematics and physics;
2. derive the Euler-Lagrange equations for variational problems,
3. solve variational problems with constraints: both algebraic and isoperimetric;
4. derive conserved quantities from symmetries, and use them to solve the Euler-Lagrange equations
5. analyse the local stability of the critical points of a variational problem.
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Syllabus |
1. Motivation: brachistochrone, catenary, geodesics and minimal surfaces.
2. First variation and the Euler-Lagrange equation.
3. Isoperimetric problems.
4. Holonomic and non-holonomic constraints.
5. Newtonian, Lagrangian and Hamiltonian mechanics.
6. Noether theorem.
7. Second variation.
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Main Bibliography |
1. The Calculus of Variations, Bruce van Brunt, New York: Springer, 2004.
2. Calculus of Variations: with applications to physics and engineering. Robert Weinstock. New York: Dover, 1974.
3. Métodos Matemáticos da Mecânica Clássica. V. I. Arnold, Moscovo: Mir, 1987.
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Teaching Methodologies and Assessment Criteria |
1)Continuous Assessment (CA):
i) Two tests - 8 points each (T1, T2);
ii) Exercise resolution - 4 points (RE);
iii) If T1 + T2 + RE < 9.5 then CA = T1 + T2 + RE;
iv) If T1 + T2 + RE = 9.5, Oral Exam - 20 points (PO). In this case CA = (T1 + T2 + RE + PO)/2;
2) The student will be exempt from the final exam if the final CA grade is greater than or equal to 9.5;
3) Students will be admitted to the final exam if they obtained in CA a grade greater than or equal to 5.5;
4) If the student obtains in CA a grade less than 5.5, they will not be admitted to the final exam;
5) In the exam, all students with a grade greater than or equal to 9.5 will be called for an oral exam - 20 points. The final grade will be the arithmetic mean of the written exam and the oral exam.
6- Students with special status have their own rules defined by the academic regulations.
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Language |
Portuguese. Tutorial support is available in English.
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