Functional Example and the Euler-Lagrange Equation

Functional Derivatives can be tedious. We can simplify them by the Euler-Lagrange Equation. Here are the notes: https://raw.githubusercontent.com/Cey... For the Functional Derivatives, we introduced the Gâteaux Derivative/Variation in a previous video. This was nothing else than the generalization of a directional derivative to function spaces. It is a neat tool, but its application can be a bit tedious. We can simplify it by deriving the so-called Euler-Lagrange Equation, which is a general condition that a function inside a functional has to adhere to in order to optimize the Functional. In very simple scenarios, we can even show that the Functional Derivative equals the partial derivative of whatever is under the integral. ------- 📝 : Check out the GitHub Repository of the channel, where I upload all the handwritten notes and source-code files (contributions are very welcome): https://github.com/Ceyron/machine-lea... 📢 : Follow me on LinkedIn or Twitter for updates on the channel and other cool Machine Learning & Simulation stuff:   / felix-koehler   and   / felix_m_koehler   💸 : If you want to support my work on the channel, you can become a Patreon here:   / mlsim   ------- Timestamps: 00:00 Introduction 00:26 Defining the Lagrangian 01:31 Optimizing the Function of the Lagrangian 05:52 Integration by Parts 07:51 Fundamental Lemma 08:43 The Euler-Lagrange Equation 09:15 Example 10:23 Simplifying Functional Derivatives 12:10 Outro