Abstract
The main problem (also known as the simplest problem) of variational calculus is used to demonstrate that the theory of exact penalties can be used to find the main results of variational calculus (for example, Euler's conditions) and the tools of nonsmooth analysis can be applied to find the internal nature of the Euler equation and design new direct numerical methods for solving problems in variational calculus through the concept of the subgradient of the exact penalty function (which is essentially nonsmooth).
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