lagrangian formalism - Writing $dot{q}$ in terms of $p$ in the Hamiltonian formulation

In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of $p$? Is there any case in which we obtain for example a transcendental equation and cannot do it?

Answer

Yes, of course that the $p$-$v$ relationship may be transcendental so that it cannot be inverted in terms of elementary functions. That doesn't mean that the inverse function doesn't exist, however. Even functions that can't be written down in terms of elementary functions may exist.

For example, consider the Lagrangian $$ L =\exp(bv^2)\cdot mv^2 $$ It implies $$ p = \frac{dL}{dv} = \exp(bv^2)(2mbv^3+2mv) $$ which can't be inverted in terms of elementary functions $v=v(p)$. However, the function $v=v(p)$ still exists – although these velocity-momentum relationships aren't necessarily one-to-one in all cases.