Many of the spells that an Eldritch Knight can cast have a somatic component. If I am playing as an Eldritch Knight with a two-handed sword or weapon/shield combo, would I require the Warcaster feat?
The way I see it is that the Warcaster feat's intention was to allow those with two-handed weapons, dual wielding or sword/shield setup to cast spells, meaning that by default, Eldritch Knights shouldn't be able to cast spells with somatic components without the feat.
Best Answer
You can't cast spells while holding your two-handed sword or your sword and shield without the War Caster feat. This is spelled out on page 203 of the PHB:
However, you don't necessarily need to take the feat in order to use a two-handed sword or a sword and shield and cast spells. (I realise the previous sentence seems to contradict the first one, but I'm getting there.)
Each turn, you can interact with one object without using an action. (See page 190 of the PHB for details.) If you are using a sword and shield, this means you can, for example, sheathe your sword without using an action, then cast a spell on the same turn. Then next turn, when you want to attack with your sword, you can draw it without using an action, then attack on the same turn. Note that this leaves you unable to take opportunity attacks between these 2 turns.
You can find more information about wielding weapons and casting spells without War Caster here, here, and here.