content_image/424073.jpg content_image/424061.jpg content_image/424070.jpg content_image/424069.jpg content_image/424060.jpg content_image/424068.jpg content_image/424071.jpg content_image/424074.jpg content_image/424072.jpg content_image/424059.jpg content_image/424067.jpg
The one-form we are integrating is W ⸥ Ω = W$_{a}$ $^{(}$dq a − ( p $^{a}$/m ) dt $^{)}$, but since both q and t stay unchanged on λ , the integral is zero. As a result, to annihilate the bound- ary term for all variational fields W we are forced to chose the curves ˜ γ$_{0}$ = (˜ q$_{0}$ ( t ) , ˜ p$_{0}$ ( t ) , t ) and ˜ γ$_{1}$ = (˜ q$_{1}$ ( t ) , ˜ p$_{1}$ ( t ) , t ) in the extended phase space in such a way that their instant tangent vectors annihilate Ω at any moment t ∈ [ t$_{0}$, t$_{1}$ ] . Thus, it must hold 

or, equivalently, 

for both ˜ γ$_{0}$ and ˜ γ$_{1}$ . Here one recognizes the dynamical equations (5) as desired. 

From the boundary term analysis and the assumption about the uniqueness of the history γ$_{cl}$ which we adopted from the very beginning we can conclude that Σ$_{ext}$ ∈ U$_{(}$$_{γ}$$_{cl}$$_{,γ}$$_{cl}$$_{)}$ ⊂ U$_{γ}$$_{cl}$ . Hence Σ$_{ext}$ has the topology of a closed string attached to the classical trajectory γ$_{cl}$ . To specify its shape the second term in (A1) must be employed. The resulting variational equation, written in a coordinate- free notation, is 

This is equivalent to the following system of partial dif- ferential equations for the unknown functions q $^{a}$( t, s ) and p$_{a}$ ( t, s ) (indices a and k run from 1 , . . . , n ): 

One solution of these equations, satisfying all boundary conditions, is trivial. It is the shrunk surface Σ$_{ext}$( t, s ) = γ$_{cl}$ ( t ) . After recalling the assumption of uniqueness of γ$_{cl}$ once more, we can see that this is the only solution of (A3). If there existed a closed unshrunk extremal surface Σ$_{ext}$ , the initial set of equations of motion (5) would have at least one one-parametrical family of classical solutions between the given pair of endpoints. 

The variational principle we described above operates on a wider stringy class U than it is in fact necessary. The surface integral formula (9) requires just the restricted subset U$_{γ}$$_{cl}$ ⊂ U . The transition to U$_{γ}$$_{cl}$ is advisable for two reasons. First, in the Lagrangian case with Ω = d ( pdq − Hdt ) we obtain an equivalence (modulo additive constant) between the least action principle using S and the standard Hamilton least action principle. Explicitly, 

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Second, the wider class U contains plenty of degenerate (shrunk) surfaces, the histories ˜ γ . These are obviously stationary surfaces of S when being varied within U , since 

for any variational vector field W . However, only one of these histories, namely γ$_{cl}$ , satisfies both equations (A2) and (A3) at the same time. This fictitious problem is avoided when one works from the beginning with the stringy subclass U$_{γ}$$_{cl}$ ⊂ U . 

## Appendix B: Surface functional integral – computational details 

Let us explain here in some detail how the surface functional integral is computed. To be as tangible as possible consider a one-dimensional system only, so that the extended phase space will be the three dimensional space R $^{3}$[ q, p, t ] (more dimensions represent only a tech- nical problem). The particle is supposed to move under the combined action of the potential U ( q ) and the fric- tion force with a general A -power law. The dynamical equations are 

and in addition to them, we require that the particle satisfies the boundary conditions q ( t$_{0}$ ) = q$_{0}$ and q ( t$_{1}$ ) = q$_{1}$ . According to the definition (8), the two-form Ω is 

<img src='content_image/424068.jpg'>

Our aim is to compute the surface functional integral (9) over the stringy set U$_{γ}$$_{cl}$ = ⋃ ˜ $_{γ}$U$_{(˜}$ γ,γ$_{cl}$ $_{)}$. The direct application of (10) and (B1) together with the Stokes theorem yields 

<img src='content_image/424071.jpg'>

The nontrivial part of this is the functional integral over the stringy subset U$_{(˜}$ γ,γ$_{cl}$ $_{)}$. As mentioned earlier, Σ ∈ U$_{(˜}$ γ,γ$_{cl}$ ) is a map 

<img src='content_image/424074.jpg'>

such that for ∀ t ∈ [ t$_{0}$, t$_{1}$ ] and ∀ s ∈ [0 , 1] there holds 

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