I have a fluorescent light in a 30-year old fitting. The light has recently stopped working, though it is trying to light at each end. Replacing the tube has had no effect. However there is a little knob on the side of the fitting and when I rotate this it is breaking and reconnecting the live wire, and making the light work. I think therefore that I may have a starter problem but I am not sure. Can I replace the original big silver 'starter' (actually a capacitor) with a new starter, or is there likely to be something else wrong?
Besides the bulb, there are two other common failure points: the inductive ballast (which limits current and, for long tubes, is a step up auto-transformer) and a preheat/starter switch (not a capacitor).
Both those components are replaceable, though it may be difficult to find them. In the 1940s through the 1970s, they were available at every hardware and electrical supply store, but since standard florescent fixtures have now been completely obsolesced by compact florescent (CFL) and LEDs, compatible components are now uncommon.
If you can't find parts or they are expensive, this might be a good opportunity to replace the light fixture with a modern one. It will be about three times more efficient, quieter, more reliable, and more easily repairable. They should be quite reasonably priced too due to significant competition.
Although I agree with @wallyk, if you still prefer to fix the old fixture, then I suggest you search for a new ballast. Most of them now eliminate the need for a separate starter. Also, if you buy a quality ballast (that eliminates the need for a starter), you'll have years of reliable lighting ahead.
Apart from finding the replacement ballast, you may have some trouble figuring out how to wire it. If so, I suggest you post a picture showing the existing light's (before disconnecting anything) so we can help you figure it out.