Consider a n-dimensional manifold $M$. Suppose I have a constraint on $M$ that defines a submanifold $N\subset M$:
$$ F:M\longrightarrow \mathbb{R} $$ $$ N:=\{p\in M \ | \ F(p)=0 \} \subset M$$
The cotangent bundle $T^*M$ is a symplectic manifold with symplectic form given, in Darboux coordinates, by:
$$\omega = \sum_{i=1}^n dq_i\wedge dp_i$$
when introducing local coordinates on $M$: $(U;q_1,\cdots,q_n)$, with associated local coordinates on $T^*M$: $(T^*U;q_1,\cdots,q_n,p_1,\cdots,p_n)$
My question is simple: Is there a systematic way to write down the symplectic form on $T^*N$ in terms of the old one?