The inevitable issue between Jude and Lucy comes to a head via the Beatles song Revolution, when Jude goes to where Lucy is "working" as a "volunteer for the cause".
Lucy is the [self confessed] naive "pampered suburban chick" to use the Angela expression from American Beauty and is taken in by the "Megaphone Diplomacy/Polarization" methodology of Pablo or some-such, who Jude has correctly assessed as a "shagger".
Ironically John wrote the song after being approached by a similar person for a "protest song" so Revolution is John's refusal to do so.
John's methodology [when he felt inclined to have one] was sort of that of a "pacifist", using his art and music [and simply love] to get J Doe to think [a very hard task as Hitler described via his Mantra "It is most fortunate for governments that the people do not think"]. So Jude represents that methodology, ie he tells Lucy he doesn't have a cause, but see below.
The tiff in the office ends their friendship for the present, but it provides great leads-in for other very clever stuff from Taymor, for example a la Cranberry Sauce.