1 0 obj << 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. Variational Methods The basic problem of the calculus of variations is to determine the function that extremizes a functional. x��SMo�0���Q*E�>lE�%�n(���v(v�f �&X���GZvb��A�HQ� �����~|w �� Constraints, not necessarily linear, with their Lagrange multipliers 3. /Resources 1 0 R endobj 2 JOSE FIGUEROA-O’FARRILL Find the shortest path (i.e., geodesic) between two given points on a surface. One-dimensional problems P (u) = F (u, u ) dx, not necessarily quadratic 2. /Parent 15 0 R One-dimensional problems P(u) = R F(u;u0)dx, not necessarily quadratic 2. %�쏢 Minimization problems that can be analyzed by the calculus of variationsserve tochar- To minimize P is to solve P = 0. 5 0 obj <> {��jU���q�t�@��p�Y!��F�I�jĤ�� C��-��ۊ*,r/.��Ƈ�H�g����. All possible errors are my faults. %PDF-1.5 /MediaBox [0 0 612 792] y(x1,c)=y1. I would have included the Euler-Lagrange proof but it would have likely made the video over 15 minutes, which is a little too long for us.Questions/requests? 5 0 obj In this video, I introduce the subject of Variational Calculus/Calculus of Variations. /ProcSet [ /PDF /Text ] %���� For later videos, a background in ODEs will be needed, though you don't need to be an expert in them since I'll walk you through :)Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4M2h3MmdaMi1xTFUPatreon: https://www.patreon.com/user?u=4354534Twitter: https://twitter.com/FacultyOfKhan I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. x��YMo�F��W�H!�f�? stream stream ��(���35 �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� >> x��]I�7r���|��鵃]¾�nj'£���In�dqidϏ�L� �^�D��_T�D._. 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. 1 22 1 0 0. Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). %�쏢 6 0 obj ��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^}�/�? /Filter /FlateDecode x��V�n�6��;�(- ���*b%^�h�nkt����1�ĭ�d���̐�eY�=H�~�?$%�~��g��ϗBU��]����D %�R@y�A g�ح�����t��7�-�}�N>M'���cV�۬�9 @{%y%�_���{����z����zv��{2��*�=~q���A���PҚ�_�� �T���9{x���g��]��Ca��嵖�u^��]�#�`�����8bT^��+�TF�pz��T�Ҝ������ku��җ����o��R Time-dependent equations in which u0 = du=dt. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 10 0 R/Group<>/Tabs/S/StructParents 1>> Let me know in the comments!Prerequisites: Not many, just know Calculus 1 (obviously). 1) is free to iterate with, so that setting. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> <> ��� ���U�#h�O�q���͟��M�&�@e���ö{: ��_d�'��.&�:Y)��=��>��h�n�e(�A�}����$���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�=��)�~��� Two-dimensional problems P (u) = F (u, ux, uy) dx dy 4. * ªº¬¼f x F x y x y x dx. <> 4 0 obj 7.2. %PDF-1.5 8 0 obj Now the step will be from a nite number of variables to an in nite number. endobj Used in deriving the Euler-Lagrange equation. <>>> calculus of variations has continued to occupy center stage, witnessing major theoretical advances, along with wide-ranging applications in physics, engineering and all branches of mathematics. endobj Time-dependent equations in which u = du/dt. 13 0 obj << >> endobj That is to say Maximum and Minimum problems for functions whose domain con-tains functions, Y(x) (or Y(x1;¢¢¢x2), or n-tuples of functions). carries ordinary calculus into the calculus of variations. >> endobj XE����q�'/�Ԗ~�|v���P�.3S�S���-cq0Z�X�Rce�e�"·el�(eX;ӷhE������ޜf"`& ]?\�m�FO�I��%K�i�=P �{�J�F��~M�*�� A �O��8�Eo$�P�ؚ�=���(P�wox�����y��D^.q��Y�4r!��V�H�#no�&�/J&��, �c_���4��)�j �"ai�����m�!�n�0?�L��k�ƈ�2-��V%��Ӂ���LN���EE���g��i�Ѣ�,�~�ܢ��� G�$�B�7� In general, there can be more than one independent variable and the integrand can depend on several functions and their higher derivatives. stream 6 0 obj <> 10 0 obj F�c��&��ZD�A��Ӱ~N���L�똞kv�&��n�� ����j( >��6��6?Դ��g�R[s��L�� Ƴ$ؿ:�҄��MbF��C��yxv\՛����mbk���2�^�pUQı�Ԃ�*~��ܕ���i��s��`��Ft�0ۗ��9F�a\�: stream Thus, c(x,u(x)) = ds dt = p 1+ u′(x)2. dx dt . z3�����I2����#��ۆ��ࠇ���(���� s85��H����`a�i��t����LVZ�d:����X�$�Y�֎��GC���e�`�aN�4L��%/� �s�C�fg�[�w8��u����=^�j�e��އ5�;�&%>652�|g�. * ªº¬¼x x dx³cos. endobj 3 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. 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. 3 0 obj << ���ף�[��WOےu��=�4:勇�����;ծ��uT��J�D�T|Y��J:��,y`�x��Ւ��)-���B�re8t3�mqH�3�� ($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��� /Contents 3 0 R /Type /Page endobj 2 0 obj << There may be more to it, but that is the main point. <> <> ����z>AE������|�D�t ���A��}�eos�ל�P����sy�t'Q�B]J���jP�����_�n�*F�@�$ stream CALCULUS OF VARIATIONS c 2006 Gilbert Strang 7.2 Calculus of Variations One theme of this book is the relation of equations to minimum principles. `����h��Q�'�i�&-�(Q��-�`*ѻ�s~� � 0� �vӝ#�����h� In this video, I introduce the subject of Variational Calculus/Calculus of Variations. Constraints, not necessarily linear, with their Lagrange multipliers 3. <> /Length 1682 endstream ?�~����WW?��3��x��=@s��Ȭ*U 'XkM)��p2���f�9ŽF���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.


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