GLYPH<14> S$_{GLYPH<23>}$ GLYPH<14>GLYPH<18> = GLYPH<14> S GLYPH<14>GLYPH<18> GLYPH<0> GLYPH<23> ( GLYPH<2>$_{0}$ ) GLYPH<2>$_{GLYPH<18>}$ ( GLYPH<18>; GLYPH<16> ) = 0 ; (10) D GLYPH<2> D GLYPH<2>$_{0}$ = GLYPH<0> GLYPH<22> ( GLYPH<16> ) GLYPH<28> $^{0}$( GLYPH<2>$_{0}$ ) GLYPH<23> ( GLYPH<2>$_{0}$ ) D GLYPH<18> D GLYPH<2>$_{0}$ $_{!}$2 : (11) GLYPH<28> $^{0}$( GLYPH<2>$_{0}$ ) GLYPH<23> ( GLYPH<2>$_{0}$ ) GLYPH<17> GLYPH<0> 1 ; GLYPH<18> ( GLYPH<16> j GLYPH<2>$_{0}$ + 2 GLYPH<25> ) GLYPH<17> GLYPH<18> ( GLYPH<16> j GLYPH<2>$_{0}$ ) + 2 GLYPH<25> ; (12) GLYPH<14> S = Z 2 GLYPH<25> q 0 $^{hGLYPH<16>}$˙ GLYPH<18> GLYPH<0> @$_{ }$ GLYPH<31> GLYPH<17> GLYPH<14>  GLYPH<0> $^{GLYPH<16>}$˙ + @$_{GLYPH<18>}$ GLYPH<31> GLYPH<17> GLYPH<14>GLYPH<18> $^{i}$d GLYPH<16> : (5) GLYPH<22> ( GLYPH<16> ) = 2 6 6 4 Z 2 GLYPH<25> 0 d GLYPH<2>$_{0}$ 2 GLYPH<25> @GLYPH<18> ( GLYPH<16> j GLYPH<2>$_{0}$ ) @ GLYPH<2>$_{0}$ $_{!}$2 3 7 7 5 GLYPH<0> 1 : (13) L = ˙ GLYPH<18> 2 2 GLYPH<0> GLYPH<15> 1 X m ; n = GLYPH<0>1 V$_{m}$$_{;}$$_{n}$ exp( im GLYPH<18> GLYPH<0> in GLYPH<16> ) ; (14) GLYPH<14> S GLYPH<14>GLYPH<18> = @ L @GLYPH<18> GLYPH<0> d d GLYPH<16> @ L @ ˙ GLYPH<18> : (6) D GLYPH<18> D GLYPH<28> = GLYPH<0> 1 GLYPH<22> ( GLYPH<16> ) GLYPH<14> S GLYPH<14>GLYPH<18> ; (7) GLYPH<23> = GLYPH<15> GLYPH<23>$_{1}$ + GLYPH<15> $^{2}$GLYPH<23>$_{2}$ + : : : ; GLYPH<22> = GLYPH<15> GLYPH<22>$_{1}$ + GLYPH<15> $^{2}$GLYPH<22>$_{2}$ + : : : ; (15) GLYPH<2> = GLYPH<18> + X m ; n $^{GLYPH<16>}$GLYPH<15> GLYPH<2> (1) m ; n + : : : $^{GLYPH<17>}$exp i ( m GLYPH<18> GLYPH<0> n GLYPH<16> ) ; GLYPH<18> ( GLYPH<16> j GLYPH<2>$_{0}$ ) = GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<16> + GLYPH<2>$_{0}$ + X m ; n $^{GLYPH<16>}$GLYPH<15> GLYPH<18> (1) m ; n + GLYPH<15> $^{2}$GLYPH<18>(2) m ; n + : : : GLYPH<17> GLYPH<2> exp i $^{hGLYPH<16>}$m GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<0> n GLYPH<17> GLYPH<16> + m GLYPH<2>$_{0}$ i ; (16) GLYPH<2> GLYPH<17> GLYPH<2> ( GLYPH<18>; GLYPH<16> ) : (8) GLYPH<18> = GLYPH<18> ( GLYPH<16> j GLYPH<2>$_{0}$ ) ; (9) 5. Reconciling QFMin and Lagrangian Ghost surfaces 6. Perturbative construction of QFMin-ghost surfaces 4. Lagrangian approach Ghost curves and uncorrected QFMin curves (i.e. curves for which the auxiliary angle GLYPH<2> has been taken to be the same as the coordinate angle GLYPH<18> ) are compared in Fig. 3. For the lower-order p ; q cases the two defini- tions of almost-invariant curves are almost indistinguish- able. However, higher-order uncorrected QFMin curves can cease to be graphs over GLYPH<18> and do not agree with the isotherms found by Hudson and Breslau [1]. Thus we are led to seek a way to choose GLYPH<2> so that the two approaches may be reconciled, keeping the physical appeal and ease of implementation of the QFMin approach while retaining the superior mathematical properties of ghost surfaces. where GLYPH<2>$_{0}$ is, as yet, an arbitrary label, satisfy the Euler– Lagrange equation with GLYPH<2>$_{GLYPH<18>}$ ( GLYPH<18>; GLYPH<16> ) GLYPH<17> @ GLYPH<2> ( GLYPH<18>; GLYPH<16> ) =@GLYPH<18> . To reconcile QFMin and ghost orbits we require that the family of pseudo-orbits defined by Eqs (9) and (10) is the same family as is generated by Eq. (7). Thus the la- bels GLYPH<2>$_{0}$ and GLYPH<28> must be functionally dependent: GLYPH<28> = GLYPH<28> ( GLYPH<2>$_{0}$ ), d GLYPH<28> = GLYPH<28> $^{0}$( GLYPH<2>$_{0}$ ) d GLYPH<2>$_{0}$ . Eliminating GLYPH<14> S =GLYPH<14>GLYPH<18> between Eq. (10) and Eq. (7) and observing that GLYPH<2>$_{GLYPH<18>}$ GLYPH<17> ( D GLYPH<2> = D GLYPH<2>$_{0}$ ) = ( D GLYPH<18>= D GLYPH<2>$_{0}$ ), where D GLYPH<18>= D GLYPH<2>$_{0}$ GLYPH<17> @GLYPH<18> ( GLYPH<16> j GLYPH<2>$_{0}$ ) =@ GLYPH<2>$_{0}$ , we find the reconcilia- tion condition We first map this problem onto one of Lagrangian dy- namics in order to make better contact with the mathemat- ical literature, e.g. Ref. 5. The vector potential A = r GLYPH<18> GLYPH<0> GLYPH<31> (  ; GLYPH<18>; GLYPH<16> ) r GLYPH<16> gives our assumed form for B and leads to the action integral S [ C ] = H ( d GLYPH<18> GLYPH<0> GLYPH<31> d GLYPH<16> ), the first variation being We now define GLYPH<2>$_{0}$ so that, for all GLYPH<2>$_{0}$ , choosing GLYPH<22> ( GLYPH<16> ) so that Eq. (11) satisfies the solvability con- dition that the integral of both sides with respect to GLYPH<2>$_{0}$ over the interval [0 ; 2 GLYPH<25> ] must be 2 GLYPH<25> , giving Setting GLYPH<14> S = 0 for all GLYPH<14>  and GLYPH<14>GLYPH<18> leads to Hamiltonian equations of motion with GLYPH<18> as generalized coordinate, as canonical momentum, GLYPH<31> as Hamiltonian, and GLYPH<16> as time: ˙ GLYPH<18> = @$_{ }$ GLYPH<31> and ˙ = GLYPH<0> @$_{GLYPH<18>}$ GLYPH<31> . The transition to the Lagrangian approach is made in the usual way, by solving ˙ GLYPH<18> = @$_{ }$ GLYPH<31> to give as a function of ˙ GLYPH<18> and writing the action as S = $^{R}$L d GLYPH<16> , where L ( GLYPH<18>; ˙ GLYPH<18> ; GLYPH<16> ) GLYPH<17> ( GLYPH<18>; ˙ GLYPH<18> ; GLYPH<16> ) ˙ GLYPH<18> GLYPH<0> GLYPH<31> (  ; GLYPH<18>; GLYPH<16> ). The first variation now becomes GLYPH<14> S = R 2 GLYPH<25> q 0 GLYPH<14>GLYPH<18> ( GLYPH<14> S =GLYPH<14>GLYPH<18> ) d GLYPH<16> , the functional derivative (action gradient) being For example, consider with the reality condition V GLYPH<3> m ; n = V$_{GLYPH<0>}$$_{m}$$_{;}$$_{GLYPH<0>}$$_{n}$ , and GLYPH<15> the expan- sion parameter. (Defining the largest perturbation coe GLYPH<14> - cient, V$_{2}$$_{;}$$_{1}$ , to be 1, GLYPH<15> = 10 GLYPH<0> 3 for the case shown in Fig. 3.) As the unperturbed system is integrable, the expan- sions of GLYPH<23> ( GLYPH<2>$_{0}$ ), GLYPH<22> ( GLYPH<16> ), and GLYPH<2> ( GLYPH<18>; GLYPH<16> ) are of the form Specialization of the general ghost-orbit gradient flow Eq. (4) to the Lagrangian form is e GLYPH<11> ected by taking P$_{ghost}$ = GLYPH<15> GLYPH<0> $^{1}$e e$_{ }$ + GLYPH<22> GLYPH<0> $^{1}$e$_{GLYPH<18>}$ e$_{GLYPH<18>}$ and taking the limit GLYPH<15> ! 0 to enforce the constraint ˙ GLYPH<18> = @$_{ }$ GLYPH<31> . Then the gradient flow becomes where GLYPH<22> ( GLYPH<16> ) = O (1) allows us to generalize the form as- sumed in Ref. 2 slightly, which we shall find necessary for reconciliation to be possible. and the ( p ; q ) QFMin pseudo-orbits are expanded as We now seek to choose GLYPH<2> so that QFMin pseudo- orbits are also the Lagrangian ghost pseudo-orbits defined above. We require [4] @$_{ }$ GLYPH<2> = 0. That is, where GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<17> p = q . At O ( GLYPH<15> ) we find GLYPH<22>$_{1}$ = 0. Also, GLYPH<2> (1) m ; n is not used in the calculation of GLYPH<18> (1) m ; $_{n}$. Thus, to first order, uncorrected ghost and QFMin pseudo-orbits are identical . This explains why the low-order uncorrected ghost and QFMin orbits appear almost identical in Fig. 3. Then GLYPH<9>$_{GLYPH<23>}$ = GLYPH<9> , L$_{GLYPH<23>}$ = L GLYPH<0> GLYPH<23> GLYPH<2> , @ L$_{GLYPH<23>}$ =@ ˙ GLYPH<18> = @ L =@ ˙ GLYPH<18> , and mem- bers of our family of QFMin pseudo-orbits