橢圓積分理論中某些術語的詞源


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在橢圓積分理論中,人們遇到與第一類不完整積分有關的術語"振幅"和"模角",這兩個變量分別表示橢圓積分的上限(振幅)和積分的某個參數(模數角)。我猜想"振幅"一詞的命名是對橢圓積分在物理問題中的起源的回憶,特別是在任意振幅擺的問題上(不在小角度的限制下)。根據這種解釋,第一類不完整橢圓積分的振幅直接對應於相應擺振動的振幅。

問題在於這種解釋與公式不符;根據幾個來源,計算橢圓周期的橢圓積分的幅度是 $ \ pi / 2 $ ,而不管鐘擺的幅度如何。

此外,我對"模塊化角度"一詞的命名一無所知,因此我也想知道其詞源。

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I don't understand French, but it seems that Legendre is the key person who coined these terms. You can use DeepL (www.deepl.com) to translate French. So if one can decipher the definition of le module then l'angle du module will begin to make sense.

Module itself is from Latin so there is no difference in French/German/ or English. Can we make sense if we take module as a "measure". This is how the unabridged Oxford Dict. tells us:

"< classical Latin modulus standard unit of measurement, rhythmic measure, pipe for controlling flow of water < modus mode n. + -ulus -ule suffix. Compare Middle French, French module (1547 in sense 3b, 1611 in Cotgrave in sense 4, 1762 in sense 6), Italian modulo (14th cent. in sense 3b, 1806 in sense 6, 1837 in sense ‘unit of measurement of running water’: compare sense 13)."

From his 1825 book, Traité des fonctions elliptiques et des intégrales Eulériennes, (available from Google books) shows the following:

Book image

"From now on we will call transcendental functions or elliptical functions, the integrals included in this formula. The transcendental H will be assumed to vanish or begin when phi=0; its extent will be determined by the variable phi, which we will call the amplitude; the constant c, always smaller than unity, will be called the modulus; we can always represent c by sin-theta, and theta will be what we call the angle of the modulus."

Translated with www.DeepL.com/Translator (free version)