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The special type of periodic motion where the restoring force is proportional to the magnitude of the displacement of the body from the mean position and acts in the direction opposite to the displacement is called simple harmonic motion.

The special type of periodic motion where the restoring force is proportional to the magnitude of the displacement of the body from the mean position and acts in the direction opposite to the displacement is called the simple harmonic motion.

I suspect that I have been following the 2nd format of writing definitions my whole life. However, now I am concerned about its validity. Which one of the two formats is correct? Should there be a "the" at the end (2nd format) or should I not write "the" (1st format)?

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    Hello, ttbas. What do raw Google searches for "the simple harmonic motion" and "simple harmonic motion" -"the simple harmonic motion" suggest? What usage does say the Wikipedia article employ? // To add a little 'clout', prominence, to the term, it's good to put it in italics the first time it's used in a definition. But no definite article here.
    – Edwin Ashworth
    Dec 21, 2022 at 12:44

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You should use the first, without the.

In English, "simple harmonic motion" is the name of that particular form of movement. That's even indicated with the verb call. English does not use an article with names when they occur simply like this.

[This] special type of periodic motion ... is called Dave.
[This] special type of periodic motion ... is called simple harmonic motion.

There are occasions when names might get an article ("the Donald"; "an Andrew"; "the London in Ontario, not England") but those are indicating a particular thing with that name, not simply naming something.

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    It's slightly more complicated, because you can't say "That thing is called chair" (it's "a chair") but I'll leave that particular nuance to someone more skilled.
    – Andrew Leach ♦
    Dec 21, 2022 at 12:49
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    It's certainly got subtleties "Dendrocopos minor is called the lesser spotted woodpecker", "the sequence of bytes at the start of a file is called the header", "Carnot's discovery is called the third law of thermodynamics", etc. Even maybe "...called the simple harmonic motion of the system".
    – Stuart F
    Dec 21, 2022 at 13:36

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