Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse

Given: The open sets $J\subseteq \mathbb{R}$, $U\subseteq {\mathbb{R}}^n$, the $C^2$ Lagrangian $\mathcal{L}$, a point $(t_0,q_0,v_0)\in J\times U\times {\mathbb{R}}^n$, and the notation ${\mathcal{L}}_{vt},{\mathcal{L}}_{vq},{\mathcal{L}}_{vv}$ for the matrices of partial derivatives of ${\mathcal{L}}_v$ with respect to the three blocks (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, $C^k$ maps and multi-index derivative notation in Euclidean space).

${\mathcal{L}}_v$ is of class $C^1$ because $\mathcal{L}$ is of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms); hence ${\mathcal{L}}_v$ is totally differentiable everywhere with derivative the linear map given by its Jacobian matrix $\left({\mathcal{L}}_{vt}{\mathcal{L}}_{vq}{\mathcal{L}}_{vv}\right)$, and that derivative depends continuously on the point (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder).

The chain rule for total derivatives (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$) computes derivatives of composites; the derivative of the identity-like map $(t,q,v)\mapsto (t,q)$ is the projection $(\tau ,\delta q,\delta v)\mapsto (\tau ,\delta q)$, and scalar multiplication of a differentiable map by $1$ is the map itself.

The Euclidean inverse function theorem (The Euclidean inverse function theorem): a $C^1$ map between open subsets of ${\mathbb{R}}^m$ whose derivative at a point is invertible has a local $C^1$ inverse there, in the sense of Continuously differentiable maps, local inverses, and local diffeomorphisms, with derivative $Dg(y)=Df(g(y){)}^{-1}$.

Linear algebra of block maps (Linear map between vector spaces over the same field, Invertible Euclidean linear maps, Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes): a linear map $L$ presented in blocks is injective iff its kernel is trivial, and on finite-dimensional spaces injectivity is equivalent to invertibility for endomorphisms: Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$ gives full image dimension from a zero kernel, hence a bijection whose inverse is linear and two-sided.