It's a matter of working in a different mathematical arena. In classical mechanics,
- Physical Space $\rightarrow $ Newtonian Formalism
- Configuration Space $\rightarrow $ Lagrangian Formalism
- Phase Space $\rightarrow $ Hamiltonian Formalism
- Hilbert Space $\rightarrow $ Koopman-von-Neumann Formalism
In Quantum Mechanics,
- Physical Space $\rightarrow $ Pilot Wave/ De Broglie-Bohm Formalism
- Configuration Space $\rightarrow $ Path Integral/ Feynman Formalism
- Phase Space $\rightarrow $ Phase-space/ Wigner-Moyal Formalism
- Hilbert Space $\rightarrow $ Wave-function/ Schrodinger Formalism
We can describe any given system -classical or quantum- using any of the mathematical arenas.
Edit: I'm not intended to give the full account of how different formalism works. But on @Andrea's comment
The two key equations of Pilot wave theory given by
$$\partial_t\rho +\nabla \left(\rho\frac{\nabla S}{m}\right)=0$$
$$\partial_t S+\frac{(\nabla S)^2}{2m}+V+Q=0$$
where $Q$ is known as the quantum potential that take account for quantum behaviors. The first equation is the usual continuity equation the second one, is completely analogous to the classical Hamilton-Jacobi equation.
The key idea is that by introduction of this new potential we get a new force on a particle, We can then calculate the trajectories of particles using Newton's usual second equation
$$m\vec{a}=\vec{F}=-\nabla (V+Q)$$
where we $V$ is normal potential and $Q$ is the quantum potential.