# calculus of variations tutorial

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N ��=m帞p �X!zF���_r�Al3����1X]��i�FL����E�Pg}2��Y���0ٿRNg���p.��Z����Kf��I�N�^,�x� j��0���SD�b��8brȘ�M�L�T�����=O�ý��sj��2r_X�ibՇc��t���ɾ��� ?�~����WW?��3��x��=@s��Ȭ*U 'XkM)��p2���f�9F���Q���3j,o.F��\$Q�_��.,&�L7��ֆ��I�g Fd���A`��V��f}���-�o f{��B�J>�)z�59t%��ɇ�]D�.h�]8�����QVa��� �;��Q�y�Ӫ�X�B����l�=k^8IU��B�Q�+lG�i �We��~�����j�NJ/�w� +�\$VR �h�O�u+x]J E�\$��x��������c�a�e�����/u��c\$��D2�:IT���\��gO�2�z��������9y�E`V@(p��L}���Ⱦ�@i=�Ę�~ؾ]uk����*�%�uds77�n2�`�De9��0\P�l��C�l���y3�r��#B�*A�2����P��FY�l�]���S��}e�כe�� ��Ķ��3z��� �H�\$�CD8�G\$��yA�X�8������)Ў��b�������^lCu��Ͳ�9`���9��vD�7}���ݡ]��-Z�Y�&2,��L�*�\$��vhW�D ��Ij��E���fE�ó�� ��!�R��7� �����?�7�~��!a���G��J� ��� ���o�݀�Cf 43ʯ�] ��q3�ߧ���-�rLj�����L\�nF� ��\$��6�l�����n����]�v�\$h b��6Q1�&�#��s���K!��C��ߡ. /Length 1682 In general, there can be more than one independent variable and the integrand can depend on several functions and their higher derivatives. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> /Length 3725 7.2. %PDF-1.5 x��WKo�F��W�q��}?ZhZ�EHc!�Z�l"�������>H�2��q���7�̮(�C�yA_x�����bqI�5-��9M3HKG�Uh�A+�G^���k�������0�M��E�p�o7ˏ`�"���/>�ݞA�jP�-a�Gg���Â0�ą�Y�z"B÷`� "տ�n���X�� �(����U���ԷU�kϼ���E ��ڃ�1Z �8f#�������?=5���I���;���QQ��O�9e�%}J?3�* �۪\G#��{竞����]�e��L�2&�Z�uC�K�z�� ~����Xޢ�}V�b����^�^ރWo몊�^'�s翘S�ۣ�)�"��u�66�k˨��������~���Ŧk5 ��Z*�S�#Vy�|���U�ŗT�����M��,2)%~WUI�\�>����Ģ��,��i��&m�|�Oqs�wձ �A�[�\$Rs�51������ a>zp��}WM\�;��z�B���Q�@��1�T�n�]�h�}i e�����K��_w�C��n���I����������!�5����+ ����E�K�Ht���"THP�0B�*}��)����P��H�2�7ES>,��P�H�6~��}�Ov#p�l|U��a��[��~c8 ���;�����[�p�}���o��v�n��b��L���v�v�܋�_Q�Os\ᒏl0��o��qJ�0e����~NBtDM4��8�0ރ�m*q�=GM�#�K�(� h%��Jo�z�j�� �t(Ϗ�,�� \:�2�'�8�ǣ�N���p6����ӡ9Ҥ�q��������#B�3R��X��L����D4y]B[�g��}����3~\$R�����.4T�SB�F��O�f��m�����E��p"�_^|O�C\3�8C�; �C���ꆢ |�3�g�c�����e��/�NW� V`KB� %���� /MediaBox [0 0 612 792] Two-dimensional problems P(u) = RR F(u;ux;uy)dxdy 4. <> %�쏢 carries ordinary calculus into the calculus of variations. 1 0 obj <> endobj endobj x��V�n�6��;�(- ���*b%^�h�nkt����1�ĭ�d���̐�eY�=H�~�?\$%�~��g��ϗBU��]����D %�R@y�A g�ح�����t��7�-�}�N>M'���cV�۬�9 stream %PDF-1.2 {��jU���q�t�@��p�Y!��F�I�jĤ�� C��-��ۊ*,r/.��Ƈ�H�g����. Constraints, not necessarily linear, with their Lagrange multipliers 3. %PDF-1.5 5 0 obj <>>> All possible errors are my faults. A�]����D}up�������0� �K��J�kּ�B�Y�5� ���8G�y.�12q;�� hipuP�k�掖�Q����a�ֱe� k��B1y� #En�"yJuR'�� +�X��-�),)Rr��"]�#I�O���ز�i��~�+KˌF��SJKk�:��Ԯ�tܴ`"з��ӥ�Qv��Rq1��b:9�kF*�bqG��W��;�. stream Two-dimensional problems P (u) = F (u, ux, uy) dx dy 4. We do it in several steps: 1. endobj >> y(c):y(x,c)x1≤ x ≤ x2(c)− δ ≤ c ≤ δ(20b) (where only the right end value ofxvaries withcsince the left end value is ﬁxed and) which saﬁsﬁes the Euler equation and. 13 0 obj << >> Our goal is to nd a function xthat minimizes the following function I(x) = Z. b a. F(t;x(t);x_(t))dt (1) where x_(t)def= dx(t) dt (2) We will nd a necessary condition for xto be at a minimum. (\$isj�� ˲+@�Jl����%)RZQ4�&-z�V�pvv�ٙ!ͮ3��:��'�_�����x����/��#N)��_eLb�ʹ2���_f�rF�h�W����b1[�ч�׍�/m��\{ 43K�u���o���X����B�/6��:�o�i3)��*~{O�\]�H�n��l9Y���_���b�(F��_���6��W;YxKqȖ��Q"����P&��Vr4��a��b�H]I]x�l,���0#�/��|�Rߩ1�y�I����ׂ���C�j~�{OMhq� ޣD�%Ёxr��'��R.�H}�C�_��}���2���a62�.��}��%~{�Q"�;ɥ?��E��� ��]�uC0K�0�h�n�P����-в*6�'���]q� K N�ѱ����H��Z���=쓩J��~��T6Z��N��0Kw`����#!�Tz#q�t�Wq�0z,�}�q{M Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. ����z>AE������|�D�t ���A��}�eos�ל�P����sy�t'Q�B]J���jP�����_�n�*F�@�\$ /Font << /F16 4 0 R /F8 5 0 R /F18 6 0 R /F19 7 0 R /F11 8 0 R /F13 9 0 R /F1 10 0 R /F10 11 0 R /F7 12 0 R /F14 13 0 R /F31 14 0 R >> Time-dependent equations in which u = du/dt. stream �C�lޘ��c�5�T+h��PT�6SD�>�)�;�L%r��s}�q4I��'CS�4:��x6yl��������0y�ӷo�}�M�G1�?��b��V� �&���~ȁSv�����b��|��"F~���/�|N���4[^�Y� <> endobj >> y(x1,c)=y1. In this video, I introduce the subject of Variational Calculus/Calculus of Variations. For a quadratic P( u) = 1 2 TKu − Tf, there is no … The Calculus of Variations is concerned with solving Extremal Problems for a Func-tional. The goal is to find a y(x) that minimizes Г, or maximizes it. %PDF-1.5 <> /Resources 1 0 R endobj ��� ���U�#h�O�q���͟��M�&�@e���ö{: ��_d�'��.&�:Y)��=��>��h�n�e(�A�}����\$���j�n��3fX�Cn����Y����앃i"�h�"A�N�=��+�Z249Gqi_��F�j��c����������|�9�|B'˞� 蘛�{��a�����W�y�q֩���� ��a���b�l/2q���]���چ��a+_]�(��Ŭ��6#�gi&RUȧ�NHo����C"W�bG1�����!��>jR�%S�=��)�~��� 8 0 obj Calculus of Variations. >> endobj A branch of mathematics that is a sort of generalization of calculus. %���� 2 0 obj The functionals dealt with in the calculus of variations are of the form. There may be more to it, but that is the main point. ]^�I15a��A��-�Ǘbx|�MHb�#g�d �� �����XX�NF� >> endobj /Type /Page Variational Methods The basic problem of the calculus of variations is to determine the function that extremizes a functional. F�c��&��ZD�A��Ӱ~N���L�똞kv�&��n�� ����j( >��6��6?Դ��g�R[s��L�� Ƴ\$ؿ:�҄��MbF��C��yxv\՛����mbk���2�^�pUQı�Ԃ�*~��ܕ���i��s��`��Ft�0ۗ��9F�a\�: <> ���ף�[��WOےu��=�4:勇�����;ծ��uT��J�D�T|Y��J:��,y`�x��Ւ��)-���B�re8t3�mqH�3�� ��(���35 (x1)+cwithy(x1,c)=y1(x1)=1 (20a) and integrating the Euler equation we get the family. endobj x��SMo�0���Q*E�>lE�%�n(���v(v�f �&X���GZvb��A�HQ� �����~|w �� endstream * ªº¬¼x x dx³cos. %���� endobj Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. 7 0 obj Minimization problems that can be analyzed by the calculus of variationsserve tochar- x��]I�7r���|��鵃]¾�ǌ'£���In�dqidϏ�L� �^�D��_T�D._. stream %PDF-1.3 Functionals. /ProcSet [ /PDF /Text ] Calculus of Variations. ��T�t��p�A��acx ����5C�"}�uM�N,�o��M#�����i��MZ�滨pmq�Oï @uc5��[)��Vj� �h�a& �x���o�&l(�@�ƅ�^\}���f���]}�ˌ^}�/�? Javier R. Movellan Copyright c 2003 Javier R. Movellan. <> endstream The condition is equivalent to the zero gradient condition for the discrete time case. The fundamental lemma of calculus of variations states that if Z b a m(x)g(x)dx= 0 (14) for all gwith continuous second partial derivatives, then m(x) = 0;for x2(a;b) (15) 3 0 obj Now the step will be from a nite number of variables to an in nite number. <> x��YMo�F��W�H!�f�? w�.̣�Tc�:WPl|n di3��꾺"�2����0�p�g���Z��8����l^{uN�MZ! 1 0 obj << Speed equals the time derivative of distance traveled, namely, the arc length of the curve y = u(x) traced by the light ray. 10 0 obj <> I describe the purpose of Variational Calculus and give some examples of problems which may be solved using techniques from Calculus of Variations.Specifically, Calculus of Variations seeks to find a function y = f(x) which makes a functional stationary.

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