[{"categories":["maths"],"content":"This is the first entry in a series on the Euler-Lagrange Equations and the Calculus of Variations. The first main result I prove is the Fundamental Lemma of the Calculus of Variations. It\u0026rsquo;s a small but important result, used not only in CoV but also in the modern theory of Partial Differential Equations, where it underpins the definition of distributional derivatives. For that reason, I want to discuss it first.\nThen I will use it to prove the Euler-Lagrange Equations in a bit more general setting than what you might find in standard textbooks.\nI write this out of personal interest, for readers with some mathematical background.\nNotation Some typical notation I will use in my blogs:\nNotation\nFor a function $f: \\mathbb{R}^n \\to \\mathbb{R},\\ (x_1, \\ldots, x_n) \\mapsto f(x)$ we denote its partial derivative with respect to $x_i$ by either of the following: $\\frac{\\partial f}{x_i}$, $\\partial_{x_i} f$, or $f_{x_i}$.\nWhat is the Calculus of Variations The theory of Calculus of Variations (CoV) is about optimizing a functional, a function that takes in another function and spits out a number. Similar to how you might want to find the point $x$ where some function $f$ has an extremum, that is, is minimized or maximized (for instance, $f(x) = {(x-1)^2}$ has minimum at $x=1$), in the CoV you want to find the input function so that the functional is minimized or maximized.\nThe typical functional for a problem in the CoV is of the form \\[ I(u) = \\int_\\Omega f(x, u(x), Du(x)) \\mathrm{d}x. \\] Here,\n$I$ is the functional whose extremum we seek; we look for a function $u$ at which functional $I(u)$ is minimal or maximal, $\\Omega \\subset \\mathbb{R}^m$ is a bounded open domain, $u: \\Omega \\to \\mathbb{R}^n$ is the input function, $Du: \\Omega \\to \\mathbb{R}^{n \\times m}$ is the Jacobian matrix of $u$, that is, \\[ Du = \\begin{pmatrix} \\frac{\\partial u_1}{\\partial x_1} \u0026 \\ldots \u0026 \\frac{\\partial u_1}{\\partial x_m} \\\\ \\vdots \u0026 \u0026 \\vdots \\\\ \\frac{\\partial u_n}{\\partial x_1} \u0026 \\ldots \u0026 \\frac{\\partial u_n}{\\partial x_m} \\end{pmatrix}, \\] $f: \\Omega \\times \\mathbb{R}^n \\times \\mathbb{R}^{n \\times m} \\to \\mathbb{R}$ is a given function used to define the functional $I$. This is a very general problem, because a function $u$ can describe many things, such as shapes, states, or processes, and a functional $I$ can represent a surface area, the action, energy, cost, to name a few. As such, this type of problem shows up everywhere, making CoV a classical and fundamental branch of mathematics.\nWhat to expect from a minimizer Remember from high school mathematics how you can try to find the minimum or maximum of a smooth function $f$ by setting its derivative to zero, i.e., $f'(x) = 0$ and solving for $x$.\n\u003c?xml version=\"1.0\" standalone=\"no\"?\u003e \u003c!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\"\u003e 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'(x)=0 In other words, if $x$ is a point where $f$ has an extremum, then $f'(x) = 0$.\nExample\nIf $f(x) = (x-1)^2$, then $f'(x) = 0$ $\\Rightarrow$ $2(x-1) = 0$, so $x = 1$. Indeed, $f$ is a parabola with minimum at $x = 1$ and $f'(1) = 0$ holds.\nThe same idea holds for multivariable functions, where it is simplest expressed using the directional derivative. The directional derivative of $f: \\mathbb R^n \\to \\mathbb R$ in the direction $v \\in \\mathbb R^n$ is given by \\[ \\nabla_v f (x) := \\frac{\\mathrm{d}}{\\mathrm d s}f(x + s \\cdot v)_{|s=0}, \\quad x \\in \\mathbb R^n. \\] \u003c?xml version=\"1.0\" standalone=\"no\"?\u003e \u003c!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\"\u003e 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+ s * v) Then the statement becomes: if $f: \\mathbb{R}^n \\to \\mathbb{R}$ is a smooth function with an extremum at the point $x \\in \\mathbb{R}^n$, then $\\nabla_v f(x) = 0$ for all $v \\in \\mathbb R^n$. In other words, \\[ \\frac{\\mathrm{d}}{\\mathrm d s}f(x + s \\cdot v)_{|s=0} = 0 \\] for any $v \\in \\mathbb{R}^n$.\nExample\nThe function $f(x) = (x_1-1)^2 + (x_2-4)^2$ describes a paraboloid with a minimum at $(1, 4)$. Checking the formula, we see that \\[ \\begin{aligned} \\nabla_v f(x) \u0026= \\frac{\\mathrm{d}}{\\mathrm d s}\\left( (x_1 + s \\cdot v_1 - 1)^2 + (x_2 + s \\cdot v_2 - 4)^2 \\right)_{|s=0} \\\\ \u0026= 2 (x_1 - 1) v_1 + 2 (x_2 - 4) v_2 = 0 \\end{aligned} \\] can only hold for all $v \\in \\mathbb R^2$ if $x = (1, 4)$, as expected.\nThis exact idea carries over to CoV, where we express it using the first variation of a functional. The first variation of a functional $I$ at $u$ in the direction $\\eta$ is given by\n\\[ \\delta I(u,\\eta) := \\frac{\\mathrm{d}}{\\mathrm d s}I(u + s \\cdot \\eta)_{|s=0}. \\]The statement for functionals then becomes: if functional $I$ attains an extremum for the function $u$, then \\[ \\delta I(u,\\eta) = \\frac{\\mathrm{d}}{\\mathrm d s}I(u + s \\cdot \\eta)_{|s=0} = 0 \\] for any smooth function $\\eta: \\Omega \\to \\mathbb R^n$ with compact support. Comparing to the multivariable case, $\\eta$ plays the role of the direction $v$.\nIntuition\nHere is another way of explaining it. I assume that $u$ is a minimizer for $I$. So, if $u$ is varied with $\\eta$, the functional $I$ should not become smaller, that is, \\[ I(u + s \\cdot \\eta) \\geq I(u) \\] for any $s$ in a neighborhood of $0$. Therefore, the derivative of $s \\mapsto I(u + s \\cdot \\eta)$ at $s=0$ should vanish.\nThe Euler-Lagrange Equation The next step is to plug in the definition of $I$. For simplicity, let\u0026rsquo;s agree on some things:\nAssume $n = 1$ so that\n$u: \\Omega \\to \\mathbb R$ its derivative can be written as $\\nabla u = {(u_{x_1}, \\ldots, u_{x_m})}$. The general case $n \u003e 1$ uses the exact same arguments applied to each component $k = 1, \\ldots, n$, which I will show later.\nAssume $f$ is smooth and $u$ is smooth.\nWrite $f(x,y,z)$, $x \\in \\Omega$, $y \\in \\mathbb R$, $z \\in \\mathbb R^{m}$ so that $f$\u0026rsquo;s partial derivatives can be written as $f_x$, $f_y$ and $\\nabla_zf = (f_{z_1}, \\ldots, f_{z_m})$.\n$C^\\infty_c(\\Omega)$ is the set of smooth functions on $\\Omega$ that have compact support in $\\Omega$ (and therefore are $0$ on the boundary $\\partial \\Omega$ of $\\Omega$).\nSo we get \\[ \\begin{aligned} 0 \u0026= \\frac{\\mathrm{d}}{\\mathrm d s}I(u + s \\cdot \\eta)_{|s=0} \\\\ \u0026= \\int_\\Omega \\frac{\\mathrm{d}}{\\mathrm d s}f(x, u(x) + s \\cdot \\eta(x), \\nabla u(x) + s \\cdot \\nabla \\eta(x))_{|s=0}\\mathrm d x \\\\ \u0026 \\qquad \\qquad \\text{(switched derivative and integral)} \\\\ \u0026= \\int_\\Omega f_y(x, u(x), \\nabla u(x)) \\eta (x) + \\nabla_z f(x, u(x), \\nabla u (x)) \\cdot \\nabla \\eta(x) \\mathrm d x \\\\ \u0026 \\qquad \\qquad \\text{(applied chain rule and set $s=0$)} \\\\ \u0026= \\int_\\Omega f_y(x, u(x), \\nabla u(x)) \\eta (x) \\mathrm d x - \\int_\\Omega \\mathrm{div}\\left( \\nabla_z f(x, u(x), \\nabla u (x)) \\right) \\eta(x) \\mathrm d x \\\\ \u0026 \\qquad \\qquad \\text{(applied integration by parts)} \\\\ \u0026= \\int_\\Omega \\left( f_y(x, u(x), \\nabla u(x)) - \\mathrm{div}\\left( \\nabla_z f(x, u(x), \\nabla u (x)) \\right) \\right) \\eta(x) \\mathrm d x \\\\ \u0026 \\qquad \\qquad \\text{(collected terms)} \\\\ \\end{aligned} \\] for any $\\eta \\in C^\\infty_c(\\Omega)$.\nThis can \u0026ldquo;clearly\u0026rdquo; only hold for every $\\eta$ if \\[ f_y(x, u(x), \\nabla u(x)) - \\mathrm{div}\\left( \\nabla_z f(x, u(x), \\nabla u (x)) \\right) = 0 \\] for all $x \\in \\Omega$. This equation is called the Euler-Lagrange Equation (ELE).\nNote\nNote that the divergence is taken of the vector field \\[ x \\mapsto \\nabla_z f(x, u(x), \\nabla u (x)), \\] as it came from the partial integration step over this vector field.\nI put the term clearly in quotes, since I made a bunch of strong assumptions on $f$ and $u$. To weaken the assumptions and make the theory of CoV richer, I will have to use a result called the Fundamental Lemma of the Calculus of Variations.\nThe Fundamental Lemma The lemma states that if a locally integrable function $f$ yields zero when tested against any test function, then $f$ must be \u0026ldquo;effectively\u0026rdquo; zero. Let me explain what I mean:\nlocally integrable means that \\[ \\int_{C} |f(x)| \\mathrm d x \u003c \\infty \\] for any compact $C \\subset \\Omega$. The space of such functions is called $L^1_{\\mathrm{loc}}(\\Omega)$. with $f$ tested against a test function $\\varphi \\in C^\\infty_c(\\Omega)$, I mean the integral $\\int_\\Omega f(x) \\varphi(x) \\mathrm d x $. with \u0026ldquo;effectively\u0026rdquo; zero I mean that $f$ is zero up to a set of measure zero. In other words, the set \\[ \\{x\\in\\Omega\\ |\\ f(x) \\neq 0\\} \\] might not be empty, but it does have measure zero. In practice, this means that $f$ is zero as far as integration is concerned: $\\int_\\Omega f(x) g(x) \\mathrm d x = 0$ for any standard function $g: \\Omega \\to \\mathbb R$ (including $g = \\mathrm{sign}(f)$ so that $\\int_\\Omega |f(x)| \\mathrm d x = 0$). In measure theory, this property is called ${ f \\equiv 0 } $ almost everywhere. With this colloquial phrasing, followed by some rigorous explanation, out of the way (as one does \u0026hellip;), let me state the result.\nLemma — Fundamental Lemma of the Calculus of Variations\nSuppose $f \\in L^1_{\\mathrm{loc}}(\\Omega)$ and \\[ \\int_\\Omega f(x)\\varphi(x) \\mathrm d x = 0 \\] for all $\\varphi \\in C^\\infty_c(\\Omega)$. Then $f \\equiv 0$ almost everywhere in $\\Omega$.\nThe proof requires some knowledge on mollifiers, which is outside the scope of this blog post. For details on mollifiers, see Wikipedia and Partial Differential Equations by L.C. Evans, the section in Appendix C on mollifiers.\nIn short, mollifiers are used to provide smooth approximations to any ${f \\in L^1_{\\mathrm{loc}}}$. This is needed as $f$ could be anything, such as a step function\n\\[ f(x) = \\left\\{ \\begin{aligned} 0 \u0026 \u0026 \\text{if}\\ x \u003c 0,\\\\ 1 \u0026 \u0026 \\text{if}\\ x \\geq 0, \\end{aligned} \\right. \\] which is not a nice smooth function.\nFor now, assume that a standard mollifier $\\varphi_\\varepsilon$ exists, that is, a family of smooth compactly supported functions with the properties:\n$\\varphi_\\varepsilon$ is supported in $B_\\varepsilon(0) := \\{x \\in \\mathbb R^n\\ |\\ |x|\u003c\\varepsilon \\}$ $f_\\varepsilon(x) := \\int_\\Omega f(y) \\varphi_\\varepsilon(x-y) \\mathrm d y$ is well-defined and approximates $f$ as follows: $f_\\varepsilon(x) \\to f(x)$ as $\\varepsilon \\to 0$ for almost every $x \\in \\Omega$ (the set where the limit does not hold has measure zero). Proof — Fundamental Lemma of the Calculus of Variations\nNote that the integral $\\int_\\Omega f \\varphi$ is well-defined for any $\\varphi \\in C^\\infty_c(\\Omega)$, as \\[ \\int_\\Omega |f \\varphi| \\leq || \\varphi||_\\infty \\int_{\\mathrm{spt}(\\varphi)} |f| \u003c \\infty, \\] since $\\mathrm{spt}(\\varphi) := \\overline{\\{x\\in\\Omega\\ |\\ \\varphi(x) \\neq 0 \\}}$ is compact in $\\Omega$.\nLet $\\varphi_\\varepsilon$ be the standard mollifier. Fix $x \\in \\Omega$ and let $\\varepsilon \u003e 0$ be small enough such that $\\overline{B_\\varepsilon(x)}:= \\{ y \\in \\mathbb R^n\\ |\\ |y - x|\\leq \\varepsilon\\} \\subset \\Omega$. Then the function $y\\mapsto \\varphi_\\varepsilon(x-y)$ is in $C^\\infty_c(\\Omega)$, since its support lies in $B_\\varepsilon(x)$.\nSo, by the assumption of the lemma, for all such $\\varepsilon$, \\[ f_\\varepsilon (x) = \\int_\\Omega f(y) \\varphi_\\varepsilon(x-y) \\mathrm d y = 0. \\] Now, by the property of mollifiers, we know that $f_\\varepsilon(x) \\to f(x)$ as $\\varepsilon \\to 0$ for almost all $x\\in\\Omega$, hence for almost all $x \\in \\Omega$ we know that $f(x)$ is the limit of zeroes. It follows that $f(x) = 0$ for almost all $x \\in \\Omega$.\nThe Classical Theorem Let me very precisely formulate the conditions, so that I can state the classical theorem involving the Euler-Lagrange Equations in the generality stated at the start of this post. In particular, we do not assume $n = 1$, that is, we look for a function ${u: \\mathbb R^m \\supset \\Omega \\to \\mathbb R^n}$.\nAssume that:\n$\\Omega\\subset \\mathbb R^m$ is open and bounded with a $C^1$-boundary $\\partial \\Omega$. $f: \\overline{\\Omega} \\times \\mathbb R^n \\times \\mathbb R^{n \\times m} \\to \\mathbb R, (x,y,z) \\mapsto f(x,y,z)$ is twice continuously differentiable. $g: \\partial \\Omega \\to \\mathbb R^n$ is a given $C^1$-function. The goal is to find a minimizer or maximizer $u$ in the class of functions \\[ \\Phi := \\left\\{ u \\in C^1(\\overline{\\Omega}; \\mathbb R^n)\\ |\\ u = g\\ \\text{on}\\ \\partial \\Omega \\right\\}. \\] From this it is clear that $g$ represents the boundary condition we impose on the solution $u$.\nI need to introduce some bookkeeping for the derivatives of $f$, to keep the formulas simple:\nThe derivative of $f$ with respect to $y \\in \\mathbb R^n$ is given by \\[ \\nabla_y f = \\left(f_{y_1}, \\ldots, f_{y_n}\\right). \\] The derivative of $f$ with respect to $z \\in \\mathbb R^{n \\times m}$ is given by \\[ D_z f: \\overline{\\Omega} \\times \\mathbb R^n \\times \\mathbb R^{n \\times m} \\to \\mathbb{R}^{n \\times m} \\] with entries \\[ (D_z f)_{ij}(x,y,z) := f_{z_{ij}}(x,y,z). \\] I will abuse notation a bit by taking the $\\mathrm{div}$ of a matrix field as follows: \\[ \\begin{aligned} \u0026\\mathrm{div}(D_z f(x, u(x), Du(x))) := \\nabla_x \\cdot D_z f \\\\ \u0026\\quad = \\left(\\sum_{i=1}^m (f_{z_{1i}}(x, u(x), Du(x)))_{x_i}, \\ldots, \\sum_{i=1}^m (f_{z_{ni}}(x, u(x), Du(x)))_{x_i} \\right) \\end{aligned} \\] Note\nTo be absolutely clear about what I am doing here:\nI consider the matrix valued function $x \\mapsto D_z f(x, u(x), Du(x))$, I interpret it as a $n$-vector of $m$-vector fields and apply $\\mathrm{div}$ to each field component-wise. This is equivalent to interpreting $\\nabla_x$ as a $1 \\times m$ matrix and applying it to the matrix valued function.\nFinally, after all that talk and agreeing on things, we are in a position to formulate the main statement of this blog post.\nTheorem\nLet $u \\in C^2(\\overline \\Omega; \\mathbb R^n) \\cap \\Phi$ satisfy $\\delta I(u, \\varphi) = 0$ for all $\\varphi \\in C^\\infty_c(\\Omega; \\mathbb R^n)$. Then $u$ satisfies the Euler-Lagrange Equations \\[ \\nabla_y f(x, u(x), Du(x)) - \\mathrm{div}\\left(D_z f(x, u(x), Du(x))\\right) = 0 \\] for all $x \\in \\Omega$.\nProof\nFix $k \\in \\{1,\\ldots,n\\}$, pick any $\\eta \\in C^\\infty_c(\\Omega)$, and consider the test function \\[ \\varphi(x) = \\eta(x) \\cdot e_k = (0, \\ldots, 0, \\eta(x), 0, \\ldots, 0). \\] Since $\\varphi \\in C^\\infty_c(\\Omega; \\mathbb R^n)$, we know that \\[ 0 = \\delta I(u, \\varphi). \\] The computation is identical to the one above. The switch of derivative and integral is allowed due to the fact that \\[ (s,x) \\mapsto f(x, u(x) + s\\varphi(x), Du(x) + sD\\varphi(x)) \\] is $C^2$-regular on $\\overline \\Omega$ and $\\Omega$ is bounded, so its derivative with respect to $s$ at $s=0$ is a bounded function of $x$ and its integral over $\\Omega$ is also bounded.\nFollowing the steps yields, \\[ \\begin{aligned} 0 \u0026= \\int_\\Omega \\big( f_{y_k}(x,u(x),Du(x)) \\\\ \u0026\\qquad - \\sum_{i = 1}^m (f_{z_{ki}})_{x_i}(x, u(x), Du(x)) \\big) \\eta(x) \\mathrm d x. \\end{aligned} \\] Using the Fundamental Lemma of the CoV, we see that \\[ f_{y_k}(x,u(x),Du(x)) - \\sum_{i = 1}^m (f_{z_{ki}})_{x_i}(x, u(x), Du(x)) = 0 \\] for almost every $x \\in \\Omega$. However, the expression is continuous with respect to $x$, so, actually, it holds for all $x$. The identity holds for every $k$, so writing it as one vector expression yields the desired Euler-Lagrange Equations.\nSources:\nRadboud University Lecture Notes Pisa University Lecture Notes Partial Differential Equations - L.C. Evans ","permalink":"https://victorhissinkmuller.dev/posts/02_fundamental_lemma_of_cov_and_euler_lagrange_equations/","series":["Euler-Lagrange Equations and Applications"],"summary":"\u003cp\u003eThis is the first entry in a series on the Euler-Lagrange Equations and the Calculus of Variations.\nThe first main result I prove is the \u003cem\u003eFundamental Lemma of the Calculus of Variations\u003c/em\u003e.\nIt\u0026rsquo;s a small but important result, used not only in CoV but also in the modern theory of\nPartial Differential Equations, where it underpins the definition of distributional derivatives.\nFor that reason, I want to discuss it first.\u003c/p\u003e","tags":["Calculus of Variations"],"title":"Fundamental Lemma of CoV and the Euler-Lagrange Equations"},{"categories":null,"content":"I\u0026rsquo;m Victor, a developer based in the Netherlands with a Ph.D. in mathematics. My interests lie at the intersection of rigorous mathematics, theoretical physics, and software development, using Rust and C++ as my primary tools.\nThis blog is where I write about topics that interest me ranging from mathematical concepts and physical intuitions to programming techniques and computer science theory. Posts range from exploratory notes to more polished write-ups.\nAbout this site Built with Hugo and the PaperMod theme (extended with KaTex), hosted on Codeberg.\n","permalink":"https://victorhissinkmuller.dev/about/","series":null,"summary":"\u003cp\u003eI\u0026rsquo;m Victor, a developer based in the Netherlands with a Ph.D. in mathematics.\nMy interests lie at the intersection of rigorous mathematics, theoretical\nphysics, and software development, using Rust and C++ as my primary tools.\u003c/p\u003e\n\u003cp\u003eThis blog is where I write about topics that interest me ranging from mathematical\nconcepts and physical intuitions to programming techniques and computer science\ntheory. Posts range from exploratory notes to more polished write-ups.\u003c/p\u003e\n\u003ch2 id=\"about-this-site\"\u003eAbout this site\u003c/h2\u003e\n\u003cp\u003eBuilt with \u003ca href=\"https://gohugo.io\"\u003eHugo\u003c/a\u003e and the\n\u003ca href=\"https://github.com/adityatelange/hugo-PaperMod\"\u003ePaperMod\u003c/a\u003e theme (extended with \u003ca href=\"https://katex.org/\"\u003eKaTex\u003c/a\u003e), hosted on\n\u003ca href=\"https://codeberg.org\"\u003eCodeberg\u003c/a\u003e.\u003c/p\u003e","tags":null,"title":"About Me"},{"categories":[],"content":"Hello, I am Victor and in this first post I will discuss my ideas and plans for this blog.\nFirst of all, I am going to write about topics I find interesting. These will mostly be in the context of Mathematics, Physics, and Computer Science. I will look for connections between these fields, but also going in-depth in particular topics.\nThe main goal of these blog posts is two-fold:\nI can write about whatever I happen to be learning at that time, so that the new material will stick. Topics I am knowledgeable about I can discuss in-depth and expose what I find interesting. The target audience will differ per topic. Stuff that is new to me should target entry level readers. Mathematics I am knowledgeable about could target graduate level math readers.\nMy first series will be on the Calculus of Variations and where it shows up. The plan is to derive the Euler-Lagrange Equations and then explore their applications in:\nNewtonian physics, leading to the Lagrangian and Hamiltonian equations of motion (and, perhaps, a follow-up introducing Poisson Geometry); Differential Geometry and Special Relativity, by way of deriving the Geodesic Equations. Further down the line, I hope to go into some computer science topics as well, maybe something on C++ or Rust. For instance, I would love to demystify move semantics in C++ in full detail in a clear way.\nTo anyone reading along: I hope you enjoy these posts.\nCheers!\n","permalink":"https://victorhissinkmuller.dev/posts/01_hello_this_blog/","series":null,"summary":"\u003cp\u003eHello, I am Victor and in this first post I will discuss my ideas and plans for this\nblog.\u003c/p\u003e\n\u003cp\u003eFirst of all, I am going to write about topics I find interesting. These will mostly\nbe in the context of Mathematics, Physics, and Computer Science. I will look for\nconnections between these fields, but also going in-depth in particular topics.\u003c/p\u003e\n\u003cp\u003eThe main goal of these blog posts is two-fold:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eI can write about whatever I happen to be learning at that time, so that the new\nmaterial will stick.\u003c/li\u003e\n\u003cli\u003eTopics I am knowledgeable about I can discuss in-depth and expose what I find\ninteresting.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThe target audience will differ per topic. Stuff that is new to me should target\nentry level readers. Mathematics I am knowledgeable about could target graduate level\nmath readers.\u003c/p\u003e","tags":["blog"],"title":"Hello this blog"},{"categories":null,"content":"","permalink":"https://victorhissinkmuller.dev/topics/","series":null,"summary":"","tags":null,"title":"Topics"}]