In the first chapter of Griffiths' Exterior Differential Systems and the Calculus of Variations, he discusses a bit of the language of jet manifolds. Namely, for a smooth manifold $M$, he considers the manifold of $k$-jets of curves, $J^k(\Bbb R,M)$. Say the the coordinates in $J^k(\Bbb R,M)$ induced by coordinates $(y^\alpha)$ in $M$ are $(x;\dot{y}^\alpha; \ddot{y}^\alpha;\ldots; (y^\alpha)^{(k)})$.
He claims that for every $k$ there is a canonical subbundle $W^*\subseteq T^*(J^k(\Bbb R,M))$, which he proceeds to describe in the case $k=1$ by using the projection $\pi: J^1(\Bbb R,M)\cong \Bbb R\times TM \to \Bbb R\times M$, given by $\pi(x,p,v)=(x,p)$ as follows: since $T_{(x,p)}(\Bbb R\times M)=T_x\Bbb R\oplus T_pM$, we have that $\partial_x|_x+v$ is there. So consider the annihilator $(\partial_x|_x+v)^\perp\subseteq T_{(x,p)}^*(\Bbb R\times M)$, and pull that back to define $$W^*_{(x,p,v)}=\pi^*\big((\partial_x|_x+v)^\perp\big).$$Then it is not hard to check that $W^*$ is locally spanned by $\theta^\alpha = {\rm d}y^\alpha-\dot{y}^\alpha {\rm d}x$. After this he describes in a complicated way how to do this recursively for general $k$. Questions:
(1) Why particularize everything for $J^k(\Bbb R,M)$? Why not look at $J^k(M,N)$?
(2) What is the significance of this $W^*$? Why should we care about it?
(3) in the kernel of $\theta^\alpha$, we'd have the relation ${\rm d}y^\alpha/{\rm d}x = \dot{y}^\alpha$, which I feel that should hold "anyway". What is the relation between this and $W^*$?
I am just starting to get used to jet manifolds, and I don't know anything about exterior differential systems yet.