등주문제에 관한 소고: 일반화된 원뿔곡선의 섭동을 활용하여 = A study on the isoperimetric problem: Using perturbations of generalized conic sections

다국어 초록 (Multilingual Abstract)

The isoperimetric problem, a long-studied and fundamental topic in mathematics, has evolved through the proofs and inquiries of many distinguished mathematicians before reaching its classical solution—the circle. Inspired by Newton’s Principia, which demonstrated that celestial bodies move in elliptical orbits under an inverse-square central force, this study defines generalized conic sections, beginning with the circle and extending through the ellipse. The properties of these multi-lobed generalized conic curves are analyzed, and isoperimetric perturbed curves are introduced by applying perturbations on the circle under the condition of equal perimeter. The areas of various isoperimetric perturbed curves are computed and compared graphically with that of the circle. The Euler–Lagrange equations are presented in detail for both unconstrained and constrained cases, and a differential-equation-based approach to the isoperimetric problem is provided. This study is expected to provide an engaging resource for learners who have studied calculus and differential equations at the undergraduate level, and to serve as a useful instructional reference for educators teaching these subjects.