Yes of course you do. As you aren't a citizen of an EU country that issues national ID cards (e.g. the Netherlands, Belgium, Germany), the only valid ID document for travelling between these countries is a passport issued by the country of which you are a citizen (while there are no systematic passport checks within Schengen, spot checks are always possible).
But in any case, you should already have it on you when departing from Brussels, given that you are legally obliged to carry your passport at all times while in Belgium - just as locals, over the age of 12, have to carry their Belgian ID/ residence cards. And while it goes beyond the remit of this particular forum, the same goes for the Netherlands (…usconsulate.gov/id/mandatory-identification…); I believe that in Germany it's not a legal requirement so if you prefer you could carry a photocopy while in that country - but on departure from/arrival in Belgium you have to be carrying the actual passport anyway.
< Do i need to carry my passport to travel on the train from Brussels to Amsterdam and Cologne. >
<< I believe that in Germany it's not a legal requirement so if you prefer you could carry a photocopy while in that country >>
As far as I know it is not strictly required in Germany, but is strongly advisable. A photocopy of a passport is useless for any official purpose, as it would be a trivial matter to falsify it.
Bring it - as Qaminari says, you are technically required to, and several friends of mine have had their passport checked on the train lately.
< you are technically required to >
It is not a "technicality" at all. It is a matter of law.
Of course you should carry it while taking trains around Europe to different countries.
Yes, you are legally required to carry your passport. You may not be asked for it. I am not a citizen of any EU or Schengen country, so I need to carry the residence permit issued to me by the country I am residing in addition to my passport. I was asked for the former only though at the hotels I have stayed in, not my passport even though I offered to show it.
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