4 a function (i.e. the forces satisfy the Helmholtz condi- tion) then we can define a local auxiliary Hamiltonian H$_{aux}$ such that f Ω = d ( p$_{a}$dq a − H$_{aux}$dt ) . Having H$_{aux}$ we can define the auxiliary classical action S$_{aux}$ . The al- ternative approach which tries to identify S$_{cl}$ with S$_{aux}$ has, however, several substantial disadvantages. First of all, the procedure of finding f (and subsequently H$_{aux}$ and S$_{aux}$ ) is highly ambiguous. Furthermore, S$_{aux}$ lacks symmetries that were originally present in the equations of motion. Because of these findings we do not follow this strategy hereinafter. From the physical point of view the dynamical equa- tions seem to be more fundamental than their com- pact but ambiguous precursors, Hamiltonian and/or La- grangian function, see [16]. Since (9) requires just the knowledge of the equations of motion, it determines the transition amplitude in a completely new way. Our proposal for the transition amplitude has ap- peared here out of thin air. Actually, we were rewriting (1) in terms of stringy surfaces and then, when realizing that the integrated function can be written without any reference to the Hamiltonian, we postulated formula (9) to be valid in general. However, this is not the whole story. The stringy functional quantization can be moti- vated also by the stringy variational principle. Having the dynamical equations and initial and final endpoints, one can form U$_{γ}$$_{cl}$ and Ω . Then using these objects one can introduce the stringy action functional GLYPH<2> ( q$_{0}$ , t$_{0}$ ) = fi x e d ( q$_{1}$ ,t$_{1}$ ) = fi x e d γ$_{cl}$ FIG. 2: Four elements (two open, one closed and one shrunk to γ$_{cl}$ ) of U$_{γ}$$_{cl}$ , anchored to the given classical history γ$_{cl}$ in the extended phase space. The two open surfaces belong to two different stringy classes U$_{(˜}$ γ,γ$_{cl}$ ) , while the closed and the shrunk surface are both elements of the same stringy class U$_{(}$$_{γ}$$_{cl}$$_{,γ}$$_{cl}$$_{)}$ . As the worldsheet Σ varies, the front and rear boundary curves, denoted in the text λ$_{0}$ and λ$_{1}$ , vary as well. This is a variational problem with varying boundaries, therefore the total variation has two terms. First one specifies the boundary and determines the initial equa- tions of motion for γ$_{cl}$ . The second one specifies the bulk of the stationary world-sheet Σ (which turns out to be shrunk into γ$_{cl}$ itself). Moreover, one immediately veri- fies that in the special case when Ω = d ( p$_{a}$dq a − Hdt ) , the stringy variational principle reduces (up to an ad- ditive constant) to the celebrated Hamilton least action principle. Of course, many subtleties were omitted here, but all of them can be found in [17], or in Appendix A. Thus, our variational principle enables us to perform the limit ℏ → 0 , in which we recover the original classical dynamics as required. formula (1) we have started with. The surface functional integral (9) is however completely free of H and requires just the knowledge of the classical equations of motion. This observation allows us to postulate (9) as a quan- tization tool in situations in which one cannot use the standard Hamiltonian approach. We just have to re- lax the requirement of the closedness of the two-form Ω , following from the definition of Ω for Hamiltonian systems. This relaxation is what is hidden behind the slightly provocative phrase one more step beyond in the title of this subsection. The problematic part here is the definition of the classi- cal action S$_{cl}$ ( q$_{1}$, t$_{1}$ | q$_{0}$, t$_{0}$ ) for non-Lagrangian systems. Later we will see that in some specific situations this quantity can be read out from the structure of the sur- face integral To examine the functionality of the proposed quanti- zation method let us first analyze the simplest friction force system. It consists of a particle with unit mass, moving in one dimension under the action of the conser- vative force F = − dU/dq and the friction force − κ ˙ q A [30]. Thus, Another possibility which comes to mind is to use the integrating factor of the generally non-closed two-form Ω . This means that one will look for a function f ( q, p, t ) on the extended phase space such that d ( f Ω) = 0 . If the structure of the dynamical equations allows for such S : U$_{γ}$$_{cl}$ → $_{R}$, Σ ↦→ S (Σ) := Σ Ω . (11) GLYPH<2> U$_{γ}$$_{cl}$ [ D Σ] exp { i ℏ GLYPH<2> Σ Ω $^{}}$. ¨ q = − κ ˙ q $^{A}$+ F ⇒ Ω = d $^{(}$pdq − 1 $_{2}$p$^{2}$dt − Udt $^{)}$− κp $^{A}$dq ∧ dt . V. QUANTIZATION OF FRICTION FORCE SYSTEMS