I am reading the work on variatonal collision integrators from the programmers point of view, since I want to implement it is the software. The work is publicly available here: caltech. I want to implement elastic collision, but I don't understand one equation, for instance, number 13:
$$ i^*\big(D_4L_d(\overline{q}_i, \tilde{\overline{q}}) + D_2L_d(\tilde{\overline{q}}, \overline{q}_{i+1})\big) = 0 $$
Where
$i^*: T^*Q \rightarrow T^*\partial C$ is the cotangent lift of the embedding $i: \partial C \rightarrow Q$.
While I have a general understanding that $i$ stands for inclusion map, since $\partial C$ is the boundary of $C \subset Q$, I completely don't understand what is cotangent lift and what it means in terms of equation.
I would be glad if someone can explain me what does it mean and, more importantly, what I can read to interpret these notations much easier without deep diving in related courses.
In general, if you have a smooth map $f:M\to N$ between two manifolds, the cotangent lift $f^*:\Omega^1(N)\to \Omega^1(M)$ is a mapping from 1-forms on $N$ to 1-forms on $M$ given by $$ (f^*\alpha)_m := \alpha_{f(m)}\circ T_mf $$ where $\alpha\in\Omega^1(N)$ is a 1-form on $N$, $m\in M$, and $T_mf:T_mM\to T_{f(m)}N$ is the derivative map of $f$ at $m$.
In your case where $i:\partial C \to Q$, the derivative map $T_ci$ is quite simple (it's just the "obvious" inclusion of $T_c\partial C$ into $T_{i(c)}Q$, where we're identifying $\partial C$ and $i(\partial C)\subset Q$), and so for $\alpha\in \Omega^1(Q)$ and $c\in\partial C$, $$ (i^*\alpha)_c = \alpha_{i(c)}\circ T_ci = \alpha_{i(c)}\vert_{T_c\partial C}. $$ That is, $(i^*\alpha)_c$ is obtained by taking the element $\alpha_{i(c)}\in T^*_{i(c)}Q$ at $i(c)$, and restricting it to $T_c\partial C$ (the tangent space to $\partial C$ at $c$).
In short, equation 13 is saying that the 1-form in brackets vanishes when applied to vectors that are tangent to the boundary $\partial C$.
Probably the best place to learn this notation is "Mechanics and Symmetry - Second Edition" by Marsden and Ratiu. Specifically Section 4.2 for the general notion of pullback of a differential form, and Section 6.3 for the notion of cotangent lift.