Why bow where we do in sh'mone esre?

Question

Based on the Bavli (B'rachos 34), Shulchan Aruch (OC 113) rules that we must bow at the start and end of the first and penultimate b'rachos of sh'mone esre (or amida). This is the general practice, too. Why those four points in sh'mone esre?

Answer 1

The Maharal in Nesiv HaAvoda ch. 10 writes (copying bits and pieces because it is lengthy):

אבל הפירוש כמו שאמרנו כי בכח הברכה הזאת הוא כל התפלה כי ברכה זאת נזכר שהוא יתברך אלהי אברהם אלהי יצחק ואלהי יעקב, והאבות הם השתלשלות העולם כי הם התחלה לעולם ובכח ההתהלה הוא הכל ובפרט ישראל אשר הם זרע האבות, ולפיכך בברכה זאת האדם קרוב אל השם יתברך ואז הוא לפניו, כי אין האדם כורע לפני המלך ואינו לפני המלך אבל כאשר מתקרב אל המלך כורע לפניו, לפיכך קודם שמזכיר השם שאז הוא קרוב אל השם יש לו לכרוע לפניו ולמסור נפשו אל השם יתברך כי אין נמצא עם השם יתברך שום מציאות והכל אפס זולתו יתברך, כי זה ענין הכריעה שהוא כורע לפניו שמוסר נפשו אליו ומבטל מציאותו אליו.‏

To summarize (with some additions I didn't copy, it's a long piece):

The first blessing includes the "force" of the entire Tefillah, because the beginning includes everything that follows it. Therefore a person is exceedingly close to Hashem in this blessing, and you bow to the King when you are in front of Him. We are acknowledging that Hashem is the source of everything, the G-d over the beginning itself. By bowing, we acknowledge that Hashem is the only existence and we nullify ourselves to Him.

וכן בסוף אותה ברכה, כי הברכה הזאת היא השתלשלות העולם מן השם יתברך כי האבות הם השתלשלות והתחלת העולם כמו שהתבאר.‏

To summarize:

So to at the end [of the first blessing], because the Avos are the beginning, as has been explained.

וכן במודים, וזה כי כשם שברכה ראשונה שהיא אבות הוא השתלשלות המציאות מן השם יתברך. וכן ברכת מודים הוראה על שכל המציאות הוא אליו.‏

To summarize:

Same with Modim, we acknowledge that all existence is Hashem's (as we say, our lives which are given into your hands etc.).

As to why it is at the beginning and end of those blessings, that is because:

ולפיכך כאשר בא אל הזכרת שמו יתברך כורע ומוסר נפשו אל השם יתברך כי מציאותו של אדם אינו נחשב אצל השם יתברך, וזוקף כאשר מזכיר שמו מפני כי מן השם יתברך הכל והוא מעמיד וזוקף הכל

To summarize:

When we come to mention of Hashem's name, we bow before it to show that our existence is nothing in context of Hashem's existence. When we reach Hashem, we straighten up because His existence gives us reality.

The idea of our bowing is to show that we are nullified to Hashem and His existence is the only real existence, as above. Hashem's name is at the beginning and end of each of those two blessings, so we bow there.

I recommend seeing the whole piece inside to understand how everything connects (plus really appetizing extensions of the idea).

Answer 2

Yearos Devash:

One should bow on 'baruch'. And generally for all the 'bowings', the revealed matter (ha-inyan nigla) is to bend one's body towards the ground because "G-d lowers the haughty and lifts up the humble". Therefore rise when saying His name, (Original Yearos Devash continues: And put to heart at this time submissiveness (hachna-a, humility to G-d), because there is to a man to look at the place where he will go as it says "For you are dust, and to dust you will return" (Bereishis 3:19). And there the body will descend and from there the soul will rise to G-d. Therefore they (the Sages) said all the while the body has not decomposed in the earth, the soul cannot rise above. And on this bowing, one directs his soul below to the ground, and from there it will rise to G-d in the (tzror hachaim) bond of eternal life. Therefore he rises afterwards when saying G-d's name...)

so it seems it is a matter of knia (submission/humility towards God)

they are in the beginning and end of the amida, as above. beginning is humility and likewise at end when you part, just like when you leave the western wall.

besides the above quote, it is logical since when you speak to a great king you must show humility before and after, how much more so to God.