What mechanisms does Trump have to force us to spend our taxpayers money and troops on something we never agreed to ?
Based on the reactions in European newspapers this past morning, it's put people on edge. Nobody seems to have this same cavalier attitude as you. Unless Europe does something quickly, it will be sidelined from the negotiations and then presented with the results as a fait accompli.
Even Ukraine is not being made party to the negotiations.
Regarding what mechanisms Trump has?
1) He could close every American base in Europe, and announce a strategic pivot to the Pacific. This may not be of much concern to Spaniards or the Irish, but it would certainly cause anxiety in places like Poland or the Baltics.
2) He could pull out of NATO entirely, and instead build a web of bilateral alliances with countries that are actually strong - like South Korea, Japan, Israel, Singapore and so on.
3) He could tariff the sh*t out of European exports to America. He's probably going to do this anyway.
4) He could designate the EU as a strategic competitor to the United States, alongside China, and restrict European access to cutting edge American innovations in semiconductor design, or other technologies.
5) As a nuclear option - Trump could sanction countries outright. He doesn't need to sanction the entire EU, just the countries that are opposed to sending forces. If you are asking what mechanisms exist - this is maybe the most effective one. Will he use it? Probably not. But would you go so far as to say it is an impossibility?
Now what can Europe do to America in return? Not much.
1) Trump constantly complains that Europe doesn't buy anything from America except for gas and oil. He's actually right about this. It also means Europe can't threaten the US by closing its markets, because it already doesn't buy anything from America.
2) If you want to spite American oil interests, the only alternative you have is to start buying oil from Russia again.
3) You could stop exporting weapons to America. This might scare them? But it would also likely kill your armaments industry since America is the number one customer.
4) You could deny the use of your ports and airports to the American military, making it much harder for them to send sh*t to the Middle East or North Africa.
5) You could place enormous export tariffs on pharma products. Currently Denmark is printing money from selling Americans weight loss drugs. If you cut their access to Ozempic and Mounjaro, maybe fat Americans will become scared. Alternatively, Novo Nordisk will just build a factory in America.