sich bücken
to bend down
To bend the upper body forward and downward, for example to pick something up from the floor.
Sie musste sich bücken, um den heruntergefallenen Stift aufzuheben.
She had to bend down to pick up the pen that had fallen.

to bend down
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to bend down
To bend the upper body forward and downward, for example to pick something up from the floor.
Sie musste sich bücken, um den heruntergefallenen Stift aufzuheben.
She had to bend down to pick up the pen that had fallen.

This verb is used almost exclusively reflexively: 'sich bücken' (to bend down). The reflexive pronoun ('mich', 'dich', 'sich', etc.) is a fixed part of it and must not be left out.
Ich bücke mich, um meine Schuhe zu binden.
I bend down to tie my shoes.
When describing the physical movement, the direction or goal is often given with a preposition. Typical constructions are 'sich nach etwas bücken' (to bend down for something) or 'sich zu etwas hinunterbücken' (to bend down toward something).
Der Gärtner bückte sich nach den reifen Erdbeeren.
The gardener bent down for the ripe strawberries.
The figurative meaning 'sich unterwerfen' (to submit) has a strongly negative connotation. It is almost always used with the preposition 'vor' (before/in front of) and a person or power in the dative.
Er war stolz und weigerte sich, sich vor dem Chef zu bücken.
He was proud and refused to bow down before the boss.
to give in / to yield
'nachgeben' and 'sich bücken' can both figuratively mean that someone follows another person's pressure. 'nachgeben' is more neutral here: you give up your resistance, perhaps in order to end an argument. 'sich bücken' is derogatory and stronger in this meaning. It means that someone behaves submissively and gives up their principles. Learners may confuse the words because both have to do with giving up. But 'sich bücken' is used only when you are clearly judging the behavior critically.

Für diese Entscheidung bückt sich der Politiker vor den mächtigen Firmen.
With this decision, the politician is bowing down to the powerful companies.
Wir geben beim Termin nicht sofort nach, sondern prüfen erst die Vorschläge.
We will not give in right away on the appointment date; instead, we will first review the proposals.