Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$ (i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q_{i},\dot{q}_{i},t)$, $i=1,2,...,n$. Generalized momenta $p_{i}$, are defined by \begin{equation} \label{7.55} p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}=p_{i}\big(q_{j},\dot{q}_{j},t\big), \ \ \ i,j=1,2,...,n \end{equation}

If the Jacobian of the generalized momenta w.r.t. the generalized velocities, or equivalently the Hessian of the Lagrangian w.r.t. the generalized velocities \begin{equation} \label{7.57} \det\Big|\frac{\partial p_{i}}{\partial\dot{q}_{j}}\Big|=\det\Big|\frac{\partial^{2}L}{\partial\dot{q}_{i}\partial\dot{q}_{j}}\Big|\neq 0 \end{equation} is non-zero, then, due to the inverse function theorem in $\mathbb{R}^{n}$, the above relations can be solved w.r.t. the generalized velocities: \begin{equation} \label{7.58} \dot{q}_{j}=\dot{q}_{j}\big(q_{i},p_{i},t\big), \ \ \ i,j=1,2,...,n \end{equation} Then, the Hamiltonian function is defined through a Legendre transform: \begin{equation} \label{7.59} H=H\big(q_{i},p_{i},t\big)=\sum_{i=1}^{n}p_{i}\dot{q}_{i}-L\big(q_{i},\dot{q}_{i},t\big) \end{equation} with $i=1,2,...,n$. The Hamtiltonian function $H:\mathbb{R}^{2n+1}\rightarrow\mathbb{R}$, is a function, of generalized coordinates, generalized momenta and time.

Now, it can be shown that, under the above assumption (i.e. the non-degeneracy of the Hessian: $\det\Big|\frac{\partial^{2}L}{\partial\dot{q}_{i}\partial\dot{q}_{j}}\Big|\neq 0$), the Lagrangian dynamical system has an equivalent Hamiltonian description.

Now the question is the following: It is frequently mentioned in the literature (but I have never seen a direct proof of this), that the above sufficient condition: $\det\Big|\frac{\partial^{2}L}{\partial\dot{q}_{i}\partial\dot{q}_{j}}\Big|\neq 0$, for a Lagrangian dynamical system to be equivalently described as a Hamiltonian system, is independent of the choice of generalized coordinates and depends only on the dynamical system itself (i.e. only on the form of the initial Lagrangian). What kind of direct proof could be provided for this ?


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