Strategies in Europe need to be different than strategies in Eastern Europe to the steppes and the middle east. Europe is mostly forest land, large rivers and cities that stayed where they are. You get cornered more easily and don’t have the ability to give ground when you are keeping a large army together, and you can back your enemy up against a river and force a fight. Europe valued heavy infantry, heavy cavalry over light mobile troops for the basic reason this is what worked in their area.
The other extreme is the steppes witch had mobile people and no real cities that stayed where they were. When you have a large area and the ability to give ground you don’t fight a stationary battle (land war in Asia thing), you give ground, make them chase you and take them when you are good and ready.
Europe did have the technological advantage. Germany produced most of the world’s steel at that time and a greater history of inventiveness with weapons and armor was centered in Europe, the Romans took the Spatha and chainmail from the ‘barbarian tribes,’ Celts, Gaul, Germanic tribes. Europe fought a less mobile, but better armored battles while the middle east and east fought a mobile, less armored fight. Part due to access to equipment part due to the area’s they lived in. A good example of tactics that worked in Europe not working in the open areas is the battle of Carrhea from the Roman times -
http://en.wikipedia.org/wiki/Battle_of_Carrhae . The crusades also had problems once they faced armies that fielded groups of horse archer as even though the armor protected them you can’t kill something you can’t catch. The peasant’s crusade was basically wiped out due to them being footmen facing mobile horse archers. They did learn to deal with it (when the armor showed up and the stallions got to fight mares in heat!!!) and were successful for a time.
As to the weapon, cavalry wants a single handed weapon, curved, strait, axe, mace or lance it needs to be a singer hander. If the grip is too long the blade becomes a problem using. A hand and a half sword is an infantry sword not a cavalry sword, thus it makes sense that a long sword is developed in an area where infantry can be better used.
Turks into Europe…One theory why the Mongols didn’t go further is the forests. The land and forests did not lend themselves to their natural way of fighting. This is just a theory though.