So, how could you keep Philadelphia as the capital of the United States? I imagine simply avoiding the mutiny or having it be resolved differently would do a lot. What changes might this make in the political situation?
Having a national government in Philadelphia or New York would do a lot in terms of slavery -- having the institution practiced in the nation's capital had a lot of impact on how it was dealt with.
So, how could you keep Philadelphia as the capital of the United States? I imagine simply avoiding the mutiny or having it be resolved differently would do a lot. What changes might this make in the political situation?
Well the mutiny makes Madison realize the importance of a Federal district and he does call for this in Federalist 43 - http://thomas.loc.gov/home/fedpapers/fed_43.html. However, also of importance was the Federal assumption of state debts. In a simple sense, this gets traded for the site of the capital.
In the War of 1812, the British don't take Washington DC, in actuality they find out its not really the city to take. Maybe they make a better effort for Baltimore, maybe go for Philadelphia which would've been a much harder blow.
In the Civil War, it also makes you wonder about what happens in Maryland. At least some pro-secessionist sentiment is quelled by the acknowledgment that the Federal government is simply not going to let it happen (and of course Lincoln suspends Habeas Corpus) and of course the fact that the capital is there does create some pro-union sentiment that might not be there if its in Philadelphia.
Its also conceivable that a Federal assumption of state debts absent a trade for the capital might even be enough to cause an early fracture of the Republic.
I can't recall why Federal assumption of state debts was disliked by Southerners? It seems like it would benefit them a lot, they have a less robust tax base than the North IIRC.
Without the DC suburbs Virginia is more Republican.