what is variable capture in nominal logic?
Variable capture is the phenomenon which "breaks" things when you do your substitutions in a naive way. For example:


Correct: in the expression $$\int_0^1 (a + x)^2 dx$$ substitute $t^2$ for $a$, to get $$\int_0^1 (t^2 + x)^2 dx.$$
Correct: in the expression $$\int_0^1 (a + t)^2 dt$$ substitute $t^2$ for $a$, to get $$\int_0^1 (t^2 + u)^2 du$$ (we renamed the bound variable $t$ to $u$ on the fly to avoid capturing $t$).
Incorrect: in the expression $$\int_0^1 (a + t)^2 dt$$ substitute $t^2$ for $a$, to get $$\int_0^1 (t^2 + t)^2 dt.$$ (We ended up with a different intergral because $t$ was captured by $dt$).


Similarly, in $\lambda$-calculus:


Correct: $(\lambda x . \lambda y . y x)) y = \lambda z . z y$
Incorrect: $(\lambda x . \lambda y . y x)) y = \lambda y . y y$   (because $y$ was captured).