
träge
sluggish
Slow, without energy or drive; heavy and listless in movement or action.
Nach dem üppigen Mittagessen fühlte er sich träge und müde.
After the large lunch, he felt sluggish and tired.
sluggish / inert
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sluggish
Slow, without energy or drive; heavy and listless in movement or action.
Nach dem üppigen Mittagessen fühlte er sich träge und müde.
After the large lunch, he felt sluggish and tired.

inert
In physics or chemistry, the property of a substance or body of only reluctantly undergoing a change of state or reacting with other substances. A synonym is 'inert'.
Edelgase wie Argon sind chemisch sehr träge und gehen kaum Verbindungen ein.
Noble gases like argon are chemically very inert and hardly form compounds.
The adjective 'träge' (sluggish/inert) belongs to the word family of 'die Trägheit' (inertia/sluggishness). This noun refers both to the human trait of lack of drive and to the physical principle of inertia.
Die Trägheit der Masse ist ein Grundprinzip der Physik.
The inertia of mass is a basic principle of physics.
You can recognize the meaning of 'träge' (sluggish/inert) from the context. If people, feelings, or slow movements are being discussed, it means 'antriebslos' (lacking drive). In scientific contexts (chemistry, physics) with words like 'Gas' (gas) or 'Stoff' (substance), it means 'does not react'.
Der Fluss bewegt sich hier sehr träge.
The river moves very sluggishly here.
In its everyday meaning, 'träge' (sluggish) is often used to describe a feeling or state. Typical phrases are 'sich träge fühlen' (to feel sluggish), 'träge werden' (to become sluggish), or also for slow, heavy movements, such as 'ein träger Fluss' (a sluggish river).
An heißen Sommertagen fühle ich mich oft besonders träge.
On hot summer days, I often feel especially sluggish.
The meaning 'inert' or 'reaktionsträge' (slow to react) is purely technical language. It is used almost exclusively in the natural sciences, such as physics and chemistry.
Stickstoff ist unter normalen Bedingungen ein relativ träges Element.
Under normal conditions, nitrogen is a relatively inert element.
tired
“Müde” and “träge” can both show a lack of energy. “Müde” refers to exhaustion or the need for rest and sleep. “Träge” describes acting slowly, without drive, or sluggishly; the person does not have to be tired.

Trotz ausreichend Schlaf bin ich heute träge und arbeite langsamer.
Despite getting enough sleep, I’m sluggish today and working more slowly.
Nach der kurzen Nacht bin ich müde und arbeite langsamer.
After the short night, I’m tired and working more slowly.