faculty of mathematics                 
                                                               and natural sciences 
                                                                                                     
                                                                
                           
                           
                           Calculus of variations and 
                                              its applications 
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
                          Bachelor Project Mathematics 
                          November 2015 
                          Student: M. H. Mudde                     
                          First supervisor: Dr. A. E. Sterk 
                          Second supervisor: Prof. dr. A. J. van der Schaft 
                          Abstract
           In this thesis, the calculus of variations is studied. We look at how opti-
           mization problems are solved using the Euler-Lagrange equation. Functions
           that satisfy this equation and the prescribed boundary conditions, must also
           satisfy Legendre’s condition and there must be no conjugate points in the
           interval in order to be the minimizer.
           Wegeneralize the Euler-Lagrange equation to higher dimensions and higher
           order derivatives to solve not only one-dimensional problems, but also multi-
           dimensional problems. At last we investigate the canonical form of the
           Euler-Lagrange equation.
           Keywords: optimization, functional, Euler-Lagrange equation, canonical
           form, Hamiltonian.
                       Acknowledgements
           I want to thank my first supervisor Dr. A. E. Sterk for being the best su-
           pervisor I could wish for. Even though I do not live in Groningen, he helped
           mein the best possible way. I could send him my drafts whenever I wanted
           and as many time I wanted, and every time he gave me adequate feedback.
           I also want to thank my second supervisor Prof. dr. A. J. van der Schaft,
           for immediately willing to be my second supervisor.
                             Contents
                             1 Introduction                                                                      3
                             2 Functions and functionals                                                         5
                                 2.1   Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       5
                                 2.2   Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . .       5
                                 2.3   Admissible functions . . . . . . . . . . . . . . . . . . . . . . .        6
                             3 Euler-Lagrange Equation                                                           8
                                 3.1   The first variation     . . . . . . . . . . . . . . . . . . . . . . . .    8
                                 3.2   Legendre transform . . . . . . . . . . . . . . . . . . . . . . . .       11
                                 3.3   Degenerate cases . . . . . . . . . . . . . . . . . . . . . . . . .       12
                                 3.4   The second variation . . . . . . . . . . . . . . . . . . . . . . .       12
                                 3.5   Necessary and sufficient conditions that determine the nature
                                       of an extremum . . . . . . . . . . . . . . . . . . . . . . . . . .       16
                             4 Applications of the Euler-Lagrange equation                                     24
                                 4.1   Curves of shortest length      . . . . . . . . . . . . . . . . . . . .   24
                                 4.2   Minimal surface of revolution . . . . . . . . . . . . . . . . . .        26
                                 4.3   The brachistochrone problem . . . . . . . . . . . . . . . . . .          29
                             5 Multi-dimensional problems                                                      32
                                 5.1   Two independent variables . . . . . . . . . . . . . . . . . . . .        32
                                 5.2   Several independent variables . . . . . . . . . . . . . . . . . .        35
                                 5.3   Several dependent variables but one independent variable . .             37
                                 5.4   Second order derivative      . . . . . . . . . . . . . . . . . . . . .   39
                                 5.5   Higher order derivatives . . . . . . . . . . . . . . . . . . . . .       42
                                 5.6   Overview of the Euler-Lagrange equations . . . . . . . . . . .           43
                             6 Applications of the multi-dimensional Euler-Lagrange equa-
                                 tions                                                                         44
                                 6.1   Minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . .       44
                                 6.2   Second order derivative problem        . . . . . . . . . . . . . . . .   46
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