8 Here, obviously, ( x$_{cl}$ ( t ) , p$_{xcl}$ ( t )) and ( y$_{cl}$ ( t ) , p$_{ycl}$ ( t )) stand for the classical solutions of (24) matching the initial and final endpoints for which the quantum transition proba- bility amplitude is sought for, x$_{cl}$ ( t$_{0}$ ) = x$_{0}$ , y$_{cl}$ ( t$_{0}$ ) = y$_{0}$ and x$_{cl}$ ( t$_{1}$ ) = x$_{1}$ , y$_{cl}$ ( t$_{1}$ ) = y$_{1}$ . In the classical limit ℏ → 0 we want the stringy propagator (9) to maintain the properties of the original dynamical system. This guides us to define: The exponential exp { i $_{ℏ}$S$_{Dcl}$ } of the classical action de- fined in such a way cancels the prefactor in (25). Finally we arrive at the following transition amplitude for the Douglas system: GLYPH<Minus> 1 1 2 3 4 5 6 GLYPH<Minus> 2 2 4 6 t$_{1}$ 〈 p 〉 0 . 6 s − 1 0 . 6 s − 1 0 . 3 s − 1 0 . 3 s − 1 0 . 1 s − 1 0 . 1 s − 1 0 . 01 s − 1 FIG. 5: Expectation value of the momentum as a function of time for an eigenstate of initial momentum Ψ$_{p}$$_{0}$( q, t$_{0}$ ) ∝ exp { i $_{ℏ}$qp$_{0}$ } , drawn for the stringy propagator (17) (blue dashed lines) and the propagator given by the Kostin- Schrödinger equation (23) (red dashed lines), respectively. The values of the friction constant κ are indicated next to the curves. The initial momentum is p$_{0}$ = 5 m s − 1 . S$_{Dcl}$ ( x$_{1}$, y$_{1}$, t$_{1}$ | x$_{0}$, y$_{0}$, t$_{0}$ ) = t$_{1}$ GLYPH<2> t$_{0}$ ( 1 2 ˙ x 2 $_{cl}$+ 1 2 ˙ y 2 cl − 1 $_{2}$y 2 cl − x$_{cl}$p$_{ycl}$ $^{)}$dt . A$_{D}$ ( x$_{1}$, y$_{1}$, t$_{1}$ | x$_{0}$, y$_{0}$, t$_{0}$ ) = 1 N$_{D}$ × ×$^{GLYPH<2>}$ [ D x D y ] exp { i ℏ t$_{1}$ GLYPH<2> t$_{0}$ ( 1 2 ˙ x 2 + 1 2 ˙ y 2 − 1 $_{2}$y 2 − xp$_{ycl}$ $^{)}$dt$^{}}$. (26) A$_{D}$ ( x$_{1}$, y$_{1}$, t$_{1}$ | x$_{0}$, y$_{0}$, t$_{0}$ ) = e i $_{ℏ}$S$_{Dcl}$ 2 πi$_{ℏ}$ ∣ ∣ ∣ ∣ ∣ ∂ $^{2}$S$_{Dcl}$ ∂x$_{1}$ ∂x$_{0}$ ∂ $^{2}$S$_{Dcl}$ ∂x$_{1}$ ∂y$_{0}$ ∂ $^{2}$S$_{Dcl}$ ∂y$_{1}$ ∂x$_{0}$ ∂ $^{2}$S$_{Dcl}$ ∂y$_{1}$ ∂y$_{0}$ ∣ ∣ ∣ ∣ ∣$^{1}$ 2 , The path integral in the configuration space we have con- structed is quadratic and hence can be computed explic- itly. The normalized transition amplitude is After we have dealt with the weak non-Lagrangeanity, let us consider the simplest strongly non-Lagrangian sys- tem. It was proposed by Jesse Douglas (one of the two winners of the first Fields Medals awarded in 1936) when studying the inverse problem of variational calculus [8]. The system is governed by the following dimensionless dynamical equations: VI. QUANTIZATION OF THE DOUGLAS SYSTEM where A quick calculation shows that the associated two-form does not possess a non-trivial local integrator f . Con- sequently, equations (24) cannot be obtained as Euler- Lagrange (Hamilton) equations. From the standard point of view, this curious situation is stalemate: no Lagrangian ⇒ no Quantum Mechan- ics . However, the surface integral method offers a way to overcome this deadlock. Using the stringy functional integration in the extended phase space we get In the paper we have developed a new quantization method that generalizes the conventional path integral approach. Throughout the paper we considered only the nonrelativistic quantum mechanics of spinless systems. However, the generalization to the field theory is rather straightforward. We have just to pass from the space of particle positions to the space of field configurations. S$_{Dcl}$ = ( x$_{1}$ − x$_{0}$ ) 2 2( t$_{1}$ t$_{0}$ ) + tan 2 $^{(}$t$_{1}$ − t$_{0}$ 2 ) ( y$_{1}$ + y$_{0}$ ) 2 2( t$_{1}$ − t$_{0}$ ) + tan ( t$_{1}$ − t$_{0}$ 2 ) ( t$_{1}$ − t$_{0}$ ) ( x$_{1}$ − x$_{0}$ )( y$_{1}$ + y$_{0}$ ) − ( x$_{1}$y$_{1}$ − x$_{0}$y$_{0}$ ) + 3 4 sin ( t$_{1}$ − t$_{0}$ )$^{[}$ ( y 2 1 + y 2 $_{0}$) cos ( t$_{1}$ − t$_{0}$ ) − 2 y$_{1}$y$_{0}$ ] − ( t$_{1}$ − t$_{0}$ ) 4 sin 2 ( t$_{1}$ − t$_{0}$ )$^{[}$ ( y 2 1 + y 2 $_{0}$) − 2 y$_{1}$y$_{0}$ cos ( t$_{1}$ − t$_{0}$ ) $^{]}$. ¨ x = − ˙ y, ¨ y = − y ⇐⇒ p$_{x}$ = ˙ x, p$_{y}$ = ˙ y, ˙ p$_{x}$ = − p$_{y}$, ˙ p$_{y}$ = y . (24) Ω = d $^{(}$p$_{x}$ dx + p$_{y}$dy − p 2 x 2 dt − p 2 y 2 dt − y 2 2 dt ) − p$_{y}$dx ∧ dt GLYPH<2> U$_{γ}$$_{cl}$ [ D Σ] exp { i ℏ GLYPH<2> Σ Ω } ∝ ∝ exp $^{{}$− i ℏ t$_{1}$ GLYPH<2> t$_{0}$ ( 1 2 ˙ x 2 cl + 1 2 ˙ y 2 cl − 1 $_{2}$y 2 cl − x$_{cl}$p$_{ycl}$ $^{)}$dt $^{}}$× ×$^{GLYPH<2>}$ [ D x D y ] exp { i ℏ t$_{1}$ GLYPH<2> t$_{0}$ ( 1 2 ˙ x 2 + 1 2 ˙ y 2 − 1 $_{2}$y 2 − xp$_{ycl}$ $^{)}$dt$^{}}$. (25) VII. CONCLUSION, DISCUSSION AND OUTLOOK The mathematical language of our exposition respects the Vladimir Arnoľd “principle of minimal generality.”