What versions of Java and Eclipse are you using, and are they 32-bit or 64-bit, and how did you install them?
Eclipse uses its own graphics library called SWT (it does not use Java AWT or Swing), and for this it needs some proprietary (non-Java) libraries. If you have 32-bit Java installed, you will need 32-bit Eclipse; if you have 64-bit Java installed, you will need 64-bit Eclipse (note that Eclipse downloads contains 32-bit and 64-bit versions for Linux).
(Note that this does not depend on whether your Linux OS is 32-bit or 64-bit, but depends on your Java).
Jesper
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