I'm trying to understand the equation on the bottom of page 71 in Robert Bryant's notes on Lie groups and symplectic geometry. It illustrates Noether's famous theorem on symmetries (or rather conservation laws) of Lagrangians so I will need some notation before posing my doubt.
Notation: Let $M$ be a smooth manifold and let $L :TM \to \mathbb{R}$ be a smooth function (Lagrangian). A curve $\gamma :[a,b]\to M$ is called $L$-critical if for any smooth variation $\Gamma_s$ of $\gamma$, we have that $0$ is a critical point of the function
$$
s \mapsto \int_a^b L\left(\frac{\partial \Gamma_s(t)}{\partial t}\right)dt
$$
If $x$ denotes a set of local coordinates on $M$, then $(x,p)$ will denote the canonically induced coordinates on $TM$. There is a $1$-form $w_L$ on $TM$ which is uniquely defined by the requirement that, locally, it should have the form
$$
w_L=\sum \frac{\partial L}{\partial p_i} dx_i.
$$
Theorem: (Noether) Let $X$ be a vector field on $M$ with local coordinates $\sum a_i(x) \frac{\partial}{\partial x_i}$ and with local flow $\alpha_t$ on $M$. Say that the Lagrangian $L$ is invariant under the derivative $d\alpha_t$, that is, $L\circ d\alpha_t = L$. Let $\widetilde{X}$ denote the vector field on $TM$ induced from the flow $d\alpha_t$. Then the function $w_L\left(\widetilde{X}\right)$ on $TM$ is constant on the curve $d\gamma$ for $\gamma :[a,b] \to M$ which is $L$-critical. $\square$
Note that, when we compute with the local coordinates $(x,p)$, we have
\begin{equation}\label{wL(X)}\tag{1}
w_L\left(\widetilde{X}\right)(x,p) = \sum \frac{\partial L(x,p)}{\partial p_i} a_i(x).
\end{equation}
Notation for Lie groups: Now we let $G$ be a Lie group and let $\{\omega_i\}$ be a basis of left-invariant $1$-forms on $G$. Let $L: TG \to \mathbb{R}$ be the Lagrangian
$$
L = (w_1)^2 + \dots + (w_n)^2.
$$
Let $\{A_i\}$ be a basis for $T_eG$ and let $Y_1$ be the right-invariant vector field on $G$ whose local flow is given by $\alpha_t(x) = \exp(tA_1)x.$ Now if $X_i$ is a left-invariant vector field given by the flow $(t,x)\mapsto x\exp(tA_i)$, then one can compute
$$
d\alpha_t(X_i(x)) (f) = X_i(x) \left(f\circ \alpha_t\right) = \frac{d}{ds}\bigg|_{s=0} f\circ \alpha_t\left( x \exp(sA_i)\right)
= \frac{d}{ds}\bigg|_{s=0} f\left(\exp(tA_1 x \exp(sA_i)\right) = X_i(\alpha_t(x))(f).
$$
Thus, $d\alpha_t(X_i) = X_i$ and so $L\circ d\alpha_t =L$ which shows that we can apply Noether's theorem with $Y_1$ as the infinitesimal symmetry. Thus the function $ w_L\left(\widetilde{Y}_1\right)$ is constant on the derivatives of $L$-critical curves.
Question: How do I see the truth of the formula
\begin{equation}\label{ques}\tag{2}
w_L\left(\widetilde{Y_1}\right) = w_1(Y_1)w_1 + w_2(Y_1) w_2 + \dots + w_n(Y_1) w_n?
\end{equation}
In particular, what coordinates do I use in formula (\ref{wL(X)}) to see formula (\ref{ques}). In general, the vector fields $X_i$ don't commute under the Lie bracket so they are not integrable to a set of coordinates.
Attempt: I'm thinking about using the chart $(x_i) \mapsto \Pi \exp(x_i A_i)$ near the identity and the induced coordinates on $TG$. Then, at least at points of the form $(0,p)$, formula (\ref{ques}) seems plausible. But I don't know what happens at other points.
Plas healp.
Best Answer
The author pointed out that this follows from the general considerations in the example preceeding my question -
We have a Lagrangian induced by a Riemannian metric. In local coordinates we can denote this by $$ L(x,p) = \sum_{i,j} g_{ij}(x) p_i p_j $$ where $[g_{ij}(x)]$ gives a symmetric nonsingular matrix for every $x$. In particular, we have a quadratic form, denoted as $G_x(\cdot,\cdot)$ on every tangent space $T_x(M)$.
Moreover, we can compute \begin{equation} \begin{split} w_L(x,p) = \sum_i \frac{\partial L(x,p)}{\partial p_i} dx_i &= \sum_i\left(2\sum_ug_{iu}(x)p_u\right)dx_i.\\ \end{split} \end{equation}
Thus, if $Y= \sum_i a_i\frac{\partial}{\partial x_i}$ is a vector field on $M$ with $\widetilde{Y}$ the associated vector field on $TM$ as written in Noether's theorem, we see that formula $(1)$ in the post becomes \begin{equation} w_L\left(\widetilde{Y}\right)(x,p) = \sum_i \frac{\partial L(x,p)}{\partial p_i} a_i(x) = 2\sum_{i u} g_{iu}(x)p_u a_i(x) = 2G_x(Y(x), p). \end{equation} That is, the $w_L\left(\widetilde{Y}\right)$ is the function on $TM$ that is obtained by simply plugging in $Y$ to one of the coordinates of the quadratic form $G$!!
Formula (2) in the post then becomes apparent (upto the constant $2$) by noting that if we use the frame of left-invariant vector fields, the matrix inducing the quadratic form (from the post) is the identity matrix.