Let $M$ be an $n$-dimensional Riemannian manifold and let $E:\Lambda M\to\mathbb{R}$ be the energy functional $E[\gamma]=\int_{S^1}||\dot\gamma(t)||^2dt$ where $\Lambda M$ is the free loop space of $M$, i.e. the manifold of loops $H^1(S^1,M)$ of Sobolev class $H^1$. Then the manifold $\Lambda M$ is a Hilbert manifold. Assume that the Riemannian metric on $M$ is chosen such that the energy functional $E$ on $\Lambda M$ is Morse-Bott. Then we know it is a $C^2$-function. Assume that it is also $C^3$ (I am not too sure if there is a way to choose this). Assume $\beta$ is a non-degenerate critical point of $E$. Then there are convex neighborhoods $V$ and $U$ of $\beta$, a diffeomorphism (of class $C^1$) $\varphi:U\to V$ with $\varphi(\beta)=\beta$ and a bounded orthogonal projection $P:\Lambda M \to \Lambda M$ such that $E[\gamma]=E[\beta]-||P(\varphi(\gamma))||_{\Lambda M}^2+||\varphi(\gamma)-P(\varphi(\gamma))||_{\Lambda M}^2$ for the orthogonal decomposition $\Lambda M=G\oplus G^{\perp}$ and the $H^1$-inner product $\langle\xi,\eta \rangle=\int_{S^1}\langle\xi,\eta\rangle+\int_{S^1}\langle D_t\xi,D_t\eta\rangle=\langle\xi,\eta\rangle_0+\langle D_t\xi,D_t\eta\rangle_0$. The dimension of the space $Im(P)$ coincides with the Morse index of the critical point $\beta$.
How do we find the projection map $P$ and the diffeomorphism $\varphi$ to express $E[\gamma]=E[\beta]-||P(\varphi(\gamma))||_{\Lambda M}^2+||\varphi(\gamma)-P(\varphi(\gamma))||_{\Lambda M}^2$ in local coordinates on $\Lambda M$?
I am looking for something similar to the ordinary Morse lemma, which asserts for $f : M\to\mathbb{R}$, there exists a chart $(x_1,x_2,...x_n)$ in a neighborhood $U$ of $\beta$ such that $x_i(\beta)=\beta$, $\forall i$ and $f(x)=f(b)-x_1^2-...-x_{\alpha}^2+x_{\alpha+1}^2+...+x_n^2$ on $U$ where $\alpha$ is equal to the index of $f$ at $\beta$.
See theorem 3 for reference.
Cross-posted on MO.
Thanks in advance!