1. STE 6239 Simulering Wednesday, Week 2: 8. More models and their simulation, Part II: variational calculus, Hamiltonians, Lagrange multipliers

2. 8.1. Introduction to the calculus of variations 1. Functions - extreme pointsFunctions - extreme points 2. Functionals - extremalsFunctionals - extremals 3. Some function spacesSome function spaces 4. Some examples of variational problemsSome examples of variational problems 5. Euler's equation for the simplest problemEuler's equation for the simplest problem 6. Two solved problemsTwo solved problems 7. Simplification of Euler´s equationSimplification of Euler´s equation 8. Proof of theorem 1Proof of theorem 1 9. Natural boundary conditionsNatural boundary conditions 10. The Euler equation for some more general casesThe Euler equation for some more general cases 11. Normed linear spacesNormed linear spaces 12. Local minimum of a functionalLocal minimum of a functional 13. Differentiation of functionalsDifferentiation of functionals 14. A necessary condition for extremum of a functionalA necessary condition for extremum of a functional 15. ExercisesExercises

3. 8.2. Introduction to Hamiltonian theory 1. The LagrangianThe Lagrangian 2. Hamilton's principleHamilton's principle 3. The HamiltonianThe Hamiltonian 4. Some examplesSome examples 5. Canonical formalismCanonical formalism 6. The general caseThe general case

4. 8.3. Equality constraints and Lagrange multipliers 1. Lagrange multipliersLagrange multipliers 2. Isoperimetric problemsIsoperimetric problems 3. Phase constraints (of type equality)

5. 8.4. More examples and exercises 1. Some more examplesSome more examples 2. ExercisesExercises

6. 8.5. From modelling to simulation: numerical approximation of solutions of variational problems 1. Approximation of integral functionals by quadratures 2. Dynamic(al) programming, with constraints 3. Example 1: numerical computation of geodesics (unconstrained)Example 1: 4. Example 2: numerical solution of an industrial problem (simulation of optimal automated steering in the presence of physical constraints)Example 2: