§0 History
The calculus of variations may be said to have begun with Newton's problem of minimal resistance, proposed in 1687, followed by the brachistochrone problem proposed by Johann Bernoulli in 1696. The subject immediately attracted the attention of Jacob Bernoulli and Marquis de l'Hôpital, but it was Leonhard Euler who began to develop it systematically in 1733. Lagrange, influenced by Euler's work, made major contributions to the theory. After Euler saw the work published in 1755 by the nineteen-year-old Lagrange, he abandoned part of his own geometric approach in favor of Lagrange's purely analytic method and renamed the subject in his 1756 lectures Elementa Calculi Variationum.
Legendre proposed a method in 1786 for distinguishing maxima from minima, although it was not entirely satisfactory. Isaac Newton and Gottfried Leibniz had also given some early attention to the subject. Other contributors included Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837). An important work by Sarrus (1842) was simplified and improved by Cauchy (1844). Other significant papers and treatises include those of Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century was that of Weierstrass. His celebrated course on the theory was epoch-making, and he may be regarded as the first person to place the calculus of variations on a firm and indisputable foundation. Hilbert's 20th and 23rd problems, published in 1900, further stimulated the development of the subject.
In the twentieth century, David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue, and Jacques Hadamard all made important contributions to the calculus of variations. Marston Morse applied variational methods to what is now known as Morse theory. Lev Pontryagin, Ralph Rockafellar, and F. H. Clarke developed new mathematical tools in optimal control theory. Richard Bellman's dynamic programming provides an alternative approach to the calculus of variations.
1 Introduction to the Upmath Project
Upmath is an open-source frontend project led by Russian particle physicist Dr. Roman Parpalak. It is an online editor based on Markdown + LaTeX. See its GitHub repository.
It can display complex mathematical formulas and graphics on the Web, including figures drawn with LaTeX TikZ.
Deploying OpenWebUI + OpenRouter API on a Debian Server with Cloudflare DNS
This article explains how to deploy a browser-accessible OpenWebUI chat interface on a Debian-based system and connect it to large language models through the OpenRouter API, creating a self-hosted ChatGPT-like web interface. We will also configure a domain using Cloudflare DNS, allowing the service to be accessed directly through
<ai.your-domain>.
This article documents two approaches for implementing one-click “中 / En” language switching in VanBlog, or in a similar blog theme with a navigation bar.
The first approach is based on the open-source project xnx3/translate, while the second uses Google Website Translator.
The main difference between the two approaches is that the open-source project uses AI translation and generally produces better translation quality, whereas Google Translate may only be accessible outside certain regions and usually provides poorer translation quality.
You can choose either approach depending on your site's performance requirements and network environment.
The article also provides the corresponding CSS, JavaScript, and HTML needed to hide translation banners, prevent the language-switching button itself from being translated, and reliably insert the switch into a specified position in the navigation bar.
Suppose we now have the domain name <your-domain>. We can use the API provided by a DNS provider, using Cloudflare as an example, to periodically update the IPv6 address.
