I would like to know if it is permitted to run on Shabbat. By this I mean run instead of walk, not as a sport. If one has to cover a long distance can one run part of it? It is physically exerting I know, but it is not work.

While there are certain restrictions on Shabbat activity, there is a special mitzvah called "Oneg Shabbat" -- which means a person should do what is enjoyable on Shabbat!

Actually, if a person is exercising for enjoyment, it is permitted to exercise on Shabbat. One should he careful, however, not to push himself too much, because to get sweaty and overexerted goes against the spirit of rest and serenity that Shabbat is designed to provide. (see "Code of Jewish Law" O.C. 301:2)