Tfillah Movements

(א) דיני הכריעות בי"ח ברכות ובו ט' סעיפים:
אלו ברכות ששוחין בהם באבות תחלה וסוף ובהודאה תחלה וסוף ואם בא לשחות בסוף כל ברכה או בתחלתה מלמדין אותו שלא ישחה אבל באמצעיתן יכול לשחות:

Shulchan Arukh, Orach Chayim 113:1

(1) (In the Shemoneh Esrei,) These are the blessings at which we bow: in Avot, at the beginning and at the end; in Modim, at the beginning and at the end. And if you come to bow at the end of every blessing or at the beginning, we teach him to not bow but in the middle, one can bow....

אורח חיים קי״ג:ד׳

(ד) המתפלל צריך שיכרע עד שיתפקקו כל חוליות שבשדרה ולא יכרע באמצע מתניו וראשו ישאר זקוף אלא גם ראשו יכוף כאגמון:

Shulchan Arukh, Orach Chayim 113:4

(4) One who is praying, he needs to bend until all the veterbre in his spine are bent. He should not bow from his hips, his head should stay straight and submissive.

אורח חיים קי״ג:ו׳

(ו) כשהוא כורע יכרע במהירות בפעם אחת וכשהוא זוקף זוקף בנחת ראשו תחלה ואחר כך גופו שלא תהא עליו כמשאוי:

Shulchan Arukh, Orach Chayim 113:6

(6) When he bows, it should be once and speedy and when he straightens up, he should straighten comfortably with his head up first and after his body so it does not look like a burden.

ברכות י״ב א:כ״ד

וְאָמַר רַבָּה בַּר חִינָּנָא [סָבָא] מִשְּׁמֵיהּ דְּרַב: הַמִּתְפַּלֵּל, כְּשֶׁהוּא כּוֹרֵעַ כּוֹרֵעַ בְּבָרוּךְ, וְכַשְּׁהוּא זוֹקֵף — זוֹקֵף בַּשֵּׁם

Berakhot 12a:24

And Rabba bar Ḥinnana Sava said in the name of Rav: One who is praying, when he bows in the appropriate places, he bows when he says: Blessed, and when he subsequently stands upright, he stands upright when he says God’s name.

ברכות ל״ד ב:א׳

כֹּהֵן גָּדוֹל, בְּסוֹף כׇּל בְּרָכָה וּבְרָכָה. וְהַמֶּלֶךְ, תְּחִלַּת כׇּל בְּרָכָה וּבְרָכָה וְסוֹף כׇּל בְּרָכָה וּבְרָכָה.

Berakhot 34b:1

It is appropriate, though, for a High Priest to bow at the end of each and every blessing; and for a king to bow at the beginning of each and every blessing and at the end of each and every blessing. This is because the more lofty one’s status, the more important it is to demonstrate his subservience to God.

ברכות ל״א א:י״ח

תַּנְיָא אָמַר רַבִּי יְהוּדָה: כָּךְ הָיָה מִנְהָגוֹ שֶׁל רַבִּי עֲקִיבָא, כְּשֶׁהָיָה מִתְפַּלֵּל עִם הַצִּיבּוּר — הָיָה מְקַצֵּר וְעוֹלֶה, מִפְּנֵי טוֹרַח צִבּוּר. וּכְשֶׁהָיָה מִתְפַּלֵּל בֵּינוֹ לְבֵין עַצְמוֹ — אָדָם מַנִּיחוֹ בְּזָוִית זוֹ, וּמוֹצְאוֹ בְּזָוִית אַחֶרֶת. וְכׇל כָּךְ לָמָּה? מִפְּנֵי כְּרִיעוֹת וְהִשְׁתַּחֲוָיוֹת.

Berakhot 31a:18

With regard to one’s intent during prayer, it was taught in a baraita that Rabbi Yehuda said: This was the custom of Rabbi Akiva, when he would pray with the congregation he would shorten his prayer and go up, due to his desire to avoid being an encumbrance on the congregation by making them wait for him to finish his prayer. But when he prayed by himself he would extend his prayers to an extent that a person would leave Rabbi Akiva alone in one corner of the study hall and later find him still praying in another corner. And why would Rabbi Akiva move about so much? Because of his bows and prostrations. Rabbi Akiva’s enthusiasm in prayer was so great, that as a result of his bows and prostrations, he would unwittingly move from one corner to the other (Rav Hai Gaon).

Passionate Prayer!
Sefer Baal Shem Tov, Chapter on Prayer 139
When you see a drowning man thrashing about, flailing his arms, trying to save himself from the churning river, you certainly will not make fun of his bizarre motions. Likewise when you see someone moving his body and gesticulating wildly when praying, don't ridicule him. He is trying to save himself from drowning in the distraction that is threatening his prayerful concentration.