Non-Standard Variational Calculus with Applications to ... - Unipa

Non-Standard Variational Calculus with Applications to Classical Mechanics 2: the inverse problem and more

F. Bagarello Dipartimento di Matematica ed Applicazioni, Fac.Ingegneria, Universit`a di Palermo, I - 90128 Palermo, Italy E-mail: [email protected]

Abstract In this paper we go on analysing the possible applications of NonStandard Analysis to variational problems, with particular interest to classical mechanics. In particular, we adapt various techniques of numerical analysis to solve the non-standard version of the Eulero-Lagrange equation, both for one- and for multi-dimensional systems. We also start an introductive analysis of the inverse problem of the calculus of variation, identifying some class of non-standard difference equations for which a first-order lagrangian can be obtained.

1

Introduction

In a previous paper, [1], we have discussed the possibility of using NonStandard Analysis (NSA in the following), [2, 3] to formulate, from the very beginning, the problem of the calculus of variation. In other words, we have used a complete non-standard approach to compute the extremum of a given R functional, J[y] ≡ ab F (x, y, y 0 ) dx, satisfying the boundary conditions y(a) = A and y(b) = B. We have successfully shown how this can be done. The action principle produce a set of ns-finite algebraic equations whose solution, at least for a large class of models, differs from the standard one, that is the one obtained by solving the usual Eulero-Lagrange differential equation related to the functional J[q], for an infinitesimal quantity. This result was obtained without finding any solution for the non-standard equations, but only by using estimates of the differences between the standard and the nonstandard solutions. The main result in [1] is, in our opinion, the criterion for the existence and the uniqueness of the solution of a differential equation which makes no longer reference to any Lipschitz condition, but only to the estimate of the lower bound of the determinant of a typically tridiagonal matrix. What was lacking in [1] is, of course, a technique for finding the explicit solution of a given non-standard difference equation. This is the first problem we will solve in this paper, at least for some class of equations. We will show that many suggestions coming from numerical analysis can be easily adapted here, and, due to the nature of NSA , they allow a different way to approach, without any approximation, the original problem. In particular we will consider some of the examples already discussed, and not solved, in [1]. In the second part we will show how to find the extremum of a functional J Rt depending on n variables, J[q1 , q2 , ..., qn ] ≡ tif L(t, q1 , q2 , ..., qn , q˙1 , q˙2 , .., q˙n ) dt and with given boundary conditions. Many examples of this problem will be discussed in details. In the last Section we will discuss some results on the inverse problem 2

of the calculus of variation. In particular, given some peculiar class of nonstandard difference equations, we will obtain the first-order lagrangian from which these equations can be derived. This paper also contains two Appendixes where we will give some information about the finite difference equations and the standard inverse problem, since we believe they are useful in order to maintain the paper self-contained.

2

Solving the Non-Standard Equations

In [1] we have discussed the non-standard version of the Eulero-Lagrange equation for a functional J[q] ≡

Z tf ti

L(t, q, q) ˙ dt,

(2.1)

where L(t, q, q) ˙ is a function with all the first and second partial derivatives continuous and the function q(t) is such that q(ti ) = qi and q(tf ) = qf . In particular we have found that the equation Lq −

d Lq˙ = 0, dt

(2.2)

in a non-standard language, must be replaced by the following set of ∆ − 1 equations: 1 qk + τ − qk−1 qk+1 − (qk + τ ) st [ (L(tk−1 , qk−1 , ) + L(tk , qk + τ, )− τ η η qk − qk−1 qk+1 − qk −L(tk−1 , qk−1 , ) − L(tk , qk , )] = 0, η η k = 1, 2, .., ∆ − 1. (2.3) We recall that our recipe consists in taking the standard part above, at a first step, with respect to τ and only at a second step also with respect to η. In [1] we have focused our attention to some existence results following from the set (2.3). In particular we have shown that it is possible to deduce 3

the existence of an unique solution of the variational problem simply by computing, or estimating, the determinant of a tridiagonal matrix. We have also discussed many classes of examples in which the solution of the system (2.3) differs from the standard solution for an infinitesimal quantity. Nevertheless we have not yet discussed how to solve the system. In this Section we want to consider in more details the finite-difference nature of (2.3), applying to this problem some techniques of this branch of mathematics. We refer to Appendix A for some information related to the standard difference equations and to some methods of solution. Let us define the shift operator (of step η) Eη as Eη f (x) ≡ f (x + η)

(2.4)

and the difference operator Dη as Dη f (x) ≡ f (x + η) − f (x),

(2.5)

so that Dη = Eη − 1. The function f (x) above belongs to a class of opportunely regular functions (for instance differentiable). In order to show the details of our procedure we consider a first order homogeneous differential equation with constant coefficients: q(t) ˙ + aq(t) = 0,

with q(ti ) = qi .

Of course this equation cannot be produced by a variational problem but, since the procedure of discretization is independent of the degree of the equation, it is still a good starting example. The standard solution is qs (t) = qi ea(ti −t) . From Appendix A and from [1] we deduce that the related non-standard equation can be obtained replacing with Dηη q . Therefore we get the time derivative dq dt (Eη + aη − 1)q(t) = 0

with q(ti ) = qi .

Let us observe that this equation gives the whole set (2.3) when η is replaced by kη, with k = 1, 2, ..., ∆ − 1. We are going to show that, whenever we 4

want to find explicitly the solution, it is sufficient to fix k = 1 and to treat the equation like a difference equation. In particular, using the results of Appendix A, we deduce that the solution of the equation above is qns (t) = qi (1 − aη)(ti −t)/η . Incidentally we observe that, since st[(1 − aη)x/η ] = e−ax for all real x, then the two solutions above belongs to the same monad for all t ∈ R , as we expected. Before considering more relevant examples, we can here discuss again the equivalence between the standard and the non-standard approaches. We have already touched this point in [1]. Now we would like to perform a deeper analysis. The point is the following: let y(x) be the (unique) solution of a given differential equation satisfying certain boundary conditions y(a) = α and y(b) = β and let {Yj } be the (unique) solution of the related nonstandard difference equation with Y0 = α and Y∆ = β. There is no reason a priori to be sure that st[y(xj ) − Yj ] = 0 for all j = 0, 1, .., ∆, even if the nonstandard difference equation ’converges’ to the standard differential equation when the standard part is considered. In the case in which this condition is satisfied we will say that the solutions converge. This peculiarity is widely discussed in the literature. For instance in [4] it is discussed that there exist approximation procedures which do not satisfy the convergence requirement even in the simple case of initial value problems for first order differential equations. Moreover, in our knowledge, there is no general result about this convergence of the solutions. What can be found in the literature, see [4, 5, 6] and many others, are only partial results related to explicit models. In these references the problem is always seen from a standard point of view, making no use of non-standard techniques. Nevertheless most of their results can be translated and used also in our approach. The most general of the equations treated in the literature, see [5, 6], is 00 y = f (x, y, y 0 ) with the boundary conditions y(a) = α and y(b) = β and with ¯ ¯ ¯ ∂f ¯ ∂f ∗ the following conditions on f : 0 < Q∗ ≤ ∂y ≤ Q and ¯ ∂y0 ¯ ≤ P ∗ . For this equation, with these hypotheses, in [5] it is shown that y(xj ) − Yj = O(h2 ), h being the discretization step. This is much more of what is needed by our 5

non-standard approach. In order to verify the convergence of the solutions it is enough to prove that y(xj )−Yj = O(hp ) for some positive real p. Below we discuss this kind of estimates for a certain number of models which are not of the form above, showing that the solutions ’converge’ and that, therefore, the two approaches are equivalent. We begin with a simple example which already does not fit into the hypotheses above. Let us consider the second order differential equation with real constant coefficients: q¨ + aq˙ + bq = 0,

q(ti ) = qi , q(tf ) = qf .

(2.6)

If b > 0 the condition above on ∂f /∂y is not satisfied so that we need some different estimate to deduce convergence of the solutions. The non-standard version of this equation, obtained as in the previous first-order example, is ³

´

Eη2 + Eη (aη − 2) + (1 − aη + bη 2 ) q(t) = 0,

(2.7) D2 q

with the same boundary conditions. The discretization of q¨ is given by ηη2 . In Appendix A we show how the general solution of such an equation can be found. The only difference is that the discretization step is here η and no longer 1. Depending on the values of a and b equations (2.6) and (2.7) admit different solutions, which must be compared with each other. Let us suppose for instance that a2 6= 4b. Therefore the general solution of the standard equation, once the boundary conditions are implemented, is (s)

(s)

qs (t) = c1 eλ1 t + c2 eλ2 t , √ √ where λ1 = 21 (−a + a2 − 4b), λ2 = 12 (−a − a2 − 4b) and (s)

c1 =

eλ2 ti qf − eλ2 tf qi eλ2 ti +λ1 tf − eλ1 ti +λ2 tf

(s)

c2 =

eλ1 tf qi − eλ1 ti qf . eλ2 ti +λ1 tf − eλ1 ti +λ2 tf

In the same hypothesis the solution of the non-standard equation is instead (ns) t/η

(ns) t/η

qns (t) = c1 q1 + c2 q2 , 6

√ √ where q1 = 12 (2 − aη + η a2 − 4b), q2 = 12 (2 − aη − η a2 − 4b) and (ns)

c1

t /η

t /η

=

q2i qf − q2f qi t /η t /η

q2i q1f

(ns)

c2

t /η t /η

− q1i q2f

=

t /η

t /η

t /η t /η

− q1i q2f

q1f qi − q1i qf q2i q1f

t /η t /η

.

What we want to control now is whether the difference qs (t) − qns (t) does or does not belong to the monad of zero for all t ∈ [ti , tf ]. This result is t/η an easy consequence of the equality st[ql ] = eλl t for l = 1, 2. To the same conclusion we arrive even for a2 = 4b, so that we can safely conclude that the solution of equation (2.6) is nothing but the standard part of the solution of equation (2.7). Let us now consider the non homogeneous equation related to (2.6), q¨ + aq˙ + bq = Φ(t),

q(ti ) = qi , q(tf ) = qf ,

(2.8)

and its non-standard version ³

´

Eη2 + Eη (aη − 2) + (1 − aη + bη 2 ) q(t) = η 2 Φ(t),

(2.9)

with the same boundary conditions. Again it is possible to show that the non-standard solution of (2.9) converges to the standard solution of (2.8), at least for a wide class of functions Φ. To prove this fact, we only need to find a particular solution of the complete equation, since the general solution of the homogeneous equation has already been found. We stress that no substantial (ns) (s) difficulty arises in the computation of the coefficients ck and ck when the function Φ is introduced. Of course, this particular solution depends on the form of Φ(t). We now consider some examples: (i) Let us suppose that Φ(t) = kht . The standard particular solution kht is qs(par) (t) = log2 h+a , while the non-standard solution is easily log h+b obtained, following the suggestions of Appendix A, and it has the form (par) (t) = ³ qns

hη −1 η

7

´2

kht +a

³

hη −1 η

´

+b

.

η

(par) Again, observing that st[ h η−1 ] = log h, we deduce that st[qns (t) − (par) qs (t)] = 0 for all t ∈ [ti , tf ], so that also the general solutions of the complete equations belong to the same monad.

(ii) let Φ(t) be a polynomial of degree n. Then the particular solutions of both (2.8) and (2.9) can be found as a polynomial of the same degree, which certainly exist if b 6= 0. These two solutions can be shown to converge to each other using some easy estimates on matrices and their determinants. It is possible, in this way, to prove that the difference between the two solutions is of O(η). The same steps can be used even if Φ(t) is a polynomial times an exponential ht . Obviously, in this case, also the solution must be searched as a polynomial times the same exponential. We notice that item (i) fits into this more general situation. (iii) If Φ(t) is a trigonometric function, eventually multiplied by a factor ht , the result follows from point (i). Let us now move to homogeneous equations of higher degree with constant coefficients. We first define two operators: L(s) =

n X k=0

ak

dn−k , dtn−k

L(ns) =

n X k=0

ak

(Eη − 1)n−k , η n−k

(2.10)

where a0 = 1. With this definition an homogeneous differential equation of degree n can be written as L(s) q(t) = 0, (2.11) while L(ns) q(t) = 0

(2.12)

is its non-standard counterpart. We know that the n independent solutions of the differential equation are obtained by the m (≤ n) solutions of its related algebraic equation n X

ak λn−k = 0,

k=0

8

(2.13)

which is the same equation that gives also the n independent solutions of the difference equation. We recall that if, for instance, λ1 is a root of (2.13) with multiplicity one, then eλ1 t and (1 + ηλ1 )t/η are respectively solutions of equations (2.11) and (2.12). We observe also that st[(1 + ηλ1 )t/η ] = eλ1 t . Similar conclusions can be obtained if the multiplicity of λ1 is bigger than one. We can state the following result: - Given the standard homogeneous differential equation and its nonstandard counterpart in (2.11) and (2.12), there exist a standard fundamental system of solution {qs(j) (t)} and a correspondent non-standard fundamental (j) (j) system of solution {qns (t)}, j = 1, ..., n, such that st|qs(j) (t) − qns (t)| = 0 for all j = 1, .., n and for all t ∈ [ti , tf ]. Up to this point we have not yet introduced the boundary conditions. If n is even, as it surely happens if the equation follows from a variational principle, see [7], we can fix the n boundary conditions on q(t) and its successive derivatives up to the order n/2. These conditions should be sufficient to fix P P (s) (ns) (i) the coefficients both in qs (t) = ni=1 ci qs(i) (t) and in qns (t) = ni=1 ci qns (t), (i) (i) where {qs } and {qns } are just the fundamental solutions introduced above. In order to conclude that qns (t) ≈ qs (t) for any time, it suffices to show (ns) (s) now that ci (t) ≈ ci (t) for all i = 1, .., n. This is actually true and can be shown using techniques analogous to the ones used for the homogeneous equations (2.6) and (2.7). The same results can also be extended to non-homogeneous equations. The convergence of the solutions is easily proved for non-homogeneousity like the ones discussed for a second order equation, with very similar techniques. Many other differential equations are under control, as long as the convergence of the solutions is concerned. We do not want to discuss these other examples here since they are related to initial value problems more than to boundary conditions ones. Therefore their relations with a variational approach is not so evident. In particular n − th order differential equations in normal form could be discussed. We plain to come back on this subject in a 9

near future. We conclude this Section by showing how two of the examples already considered in [1] can be explicitly solved using the procedure proposed in this Section. The first example is given by the differential equation q¨(t) = 1, with the 2 boundary conditions q(0) = q(1) = 0. Its standard solution is qs (t) = t 2−t . The non-standard version of q¨(t) = 1 is obviously (Eη2 − 2Eη + 1)q = η 2 . The (h) general solution of the homogeneous equation is qns (t) = c1 + c2 t, while t2 /2 is a particular solution of the complete equation. Fixing the above boundary 2 conditions we obtain the solution qns (t) = t 2−t . Therefore standard and non-standard solutions coincide! Different is the situation for the second example, q¨ = q +t, with boundary t −e−t ) conditions q(0) = q(1) = 0, whose standard solution is qs (t) = (ee−e − t. −1 2 2 2 The non-standard equation is now (Eη − 2Eη + (1 − η ))q = tη . Using the same boundary conditions we obtain now: qns (t) =

((1 + η)t/η − (1 − η)t/η ) − t. ((1 + η)1/η − (1 − η)1/η )

Recalling that st[(1 ± η)t/η ] = e±t ∀t ∈ R , we can easily verify that st[qs (t) − qns (t)] = 0 ∀t ∈ R .

3

The Eulero-Lagrange Equations in Several Variables

In this Section we will obtain the analogous of the system (2.3) for a functional J depending on more variables. Let us consider a function L(t, q1 , q2 , ..., qn , q˙1 , q˙2 , .., q˙n ), with all the first and second partial derivatives continuous. We define the functional J[q1 , q2 , ..., qn ] ≡

Z tf ti

L(t, q1 , q2 , ..., qn , q˙1 , q˙2 , .., q˙n ) dt. 10

(3.1)

The obvious generalization of the variational problem in one dimension consists now in obtaining the set of functions (q1 (t), q2 (t), ..., qn (t)), satisfying the following boundary conditions: qk (ti ) = qi,k qk (tf ) = qf,k , k = 1, 2, .., n, which are an extremum of the functional J. In this case it is quite easy to generalize the results of [1]. With the very same steps and considerations as in [1] we obtain the following set of n times ∆−1 equations: 1 qα,k + τ − qα,k−1 qα,k+1 − (qα,k + τ ) st [ (L(tk−1 , qα,k−1 , ) + L(tk , qα,k + τ, )− τ η η qα,k+1 − qα,k qα,k − qα,k−1 ) − L(tk , qα,k , ))] = 0, −L(tk−1 , qα,k−1 , η η k = 1, 2, .., ∆ − 1; α = 1, 2, .., n. (3.2) The rule for dealing with these equations is the same we have already used for the one-dimensional case: we first take the standard part with respect to τ and only after obtaining the solution also with respect to η. Of course these equations are the non-standard analogous of the well known set of n differential equations Lqα −

d Lq˙ = 0, dt α

α = 1, 2, ...., n.

(3.3)

We now consider some examples: ˙ ˙ 2 ) − 1 k(r(t) − Example 1: L(r(t), θ(t), r(t), ˙ θ(t)) = 21 m(r(t) ˙ 2 + r(t)2 θ(t) 2 r)2 + mgr(t) cos θ(t) This first example describes a particle with mass m free to move in a vertical plane, and fixed to the origin of the plane by means of an elastic string. Therefore this particle is subjected to the gravity and to the quadratic force of the string expressed by the Hooke’s law. Here r is the equilibrium position of the elastic string. We will not discuss the explicit solution of 11

the system (3.2) since even its standard counterpart is hard to manage. We will focus our attention only to the relations between the standard and the non-standard approaches. The classical Eulero-Lagrange equations are m¨ r = mrθ˙2 − k(r − r) + mg cos θ 2rr˙ θ˙ + r2 θ¨ = −gr sin θ. With algebraic calculations we get rk + τ − rk−1 θk − θk−1 rk − rk−1 θk − θk−1 , )−L(rk−1 , θk−1 , , )= η η η η " # 1 τ2 τ = m 2 + 2 2 (rk − rk−1 ) 2 η η

L(rk−1 , θk−1 ,

and L(rk + τ, θk ,

rk+1 − rk θk+1 − θk rk+1 − rk − τ θk+1 − θk , )−L(rk , θk , , )= η η η η 

Ã

θk+1 − θk 1 τ2 τ = m  2 − 2 2 (rk+1 − rk ) + (τ 2 + 2τ rk ) 2 η η η −

!2  −

1 2 kτ − kτ (rk − r) + mgτ cos θk 2

Therefore the first equation can be written as Ã

2rk − rk−1 − rk+1 θk+1 − θk m + mrk 2 η η

!2

− k(rk − r) + mg cos θk ≈ 0.

With similar calculations we get the second equation, which has the following form: rk2

θk+1 − θk rk+1 − rk 2θk − θk−1 − θk+1 − 2rk − grk sinθk ≈ 0. 2 η η η

In obtaining this equation we have taken advantage of the fact that, being by hypothesis r(t) a twice differentiable function, rk , rk−1 and rk+1 all belong to the same monad for all k. 12

We observe that, as we expected, the non-standard equations are the discretized version of the standard ones. This is in agreement with Proposition 2 of [1], which also can be restated for this multi-dimensional situation: without going into details, which are indeed very similar to those in [1], we can say that the system (3.2) is the discretized version of the standard system (3.3). Therefore, we may say that the discretization procedure commutes with the procedure of finding the variation of the functional J[q1 , q2 , .., qn ]. This means that the two procedure (a) and (b) described below give equations which, at most, differ for infinitesimal quantities: (a1) compute δJ and put δJ = 0. This gives the standard Eulero-Lagrange equations; (a2) discretize the system obtained in point (a1) using an infinitesimal time step η. (b1) discretize the functional J[q1 , q2 , .., qn ] by means of an infinitesimal time step η; (b2) compute the variation of the discretized functional J, and assume that this quantity is infinitesimal. This gives back the system (3.2). Example 2: L(q1 (t), q2 (t), q˙1 (t), q˙2 (t)) = q˙12 (t) + q˙22 (t) + 2q1 (t) + 2q2 (t) This example is really extremely simple since the lagrangian is obviously a sum of two one-body contributions and there is no ”interactions” between particle 1 and particle 2. Let us fix the following boundary conditions: qα (0) = qα (1) = 0,

α = 1, 2.

The standard solution is q1 (t) = q2 (t) = 21 (t2 − t). We will see in a moment that the same solution is found using the non-standard equations (3.2). With some algebraic computation we obtain L (q1,k−1 , q2,k−1 ,

q1,k + τ − q1,k−1 q2,k − q2,k−1 , )− η η 13

L (q1,k−1 , q2,k−1 ,

q1,k − q1,k−1 q2,k − q2,k−1 τ2 τ , ) = 2 + 2 2 (q1,k − q1,k−1 ) η η η η

and q1,k+1 − q1,k − τ q2,k+1 − q2,k , )− η η τ q1,k+1 − q1,k q2,k+1 − q2,k τ2 L (q1,k , q2,k , , ) = 2 − 2 2 (q1,k+1 − q1,k ) + 2τ, η η η η L (q1,k + τ, q2,k ,

so that, after taking the standard part with respect to τ , the first equation becomes 2q1,k − q1,k−1 − q1,k+1 = −η 2 1 ≤ k ≤ ∆ − 1. (3.4) This coincides with the non-standard equation found in Example 1 of reference [1], for the single variable qk , and it is, as discussed in this reference, the discretized version of the standard differential equation. Due to the symmetry q1 ↔ q2 of the lagrangian an equation analogous to (3.4) is obtained for q2 (t). The solution of this equation is discussed in the previous Section. We see that, considering the boundary conditions above, the solution coincides with the standard one, for all t ∈ [0, 1]. Example 3: L(q1 (t), q2 (t), q˙1 (t), q˙2 (t)) = 21 m(q˙12 (t) + q˙22 (t)) − U (q1 (t) − q2 (t)) This is actually a family of examples, depending on the explicit form of the potential U . It describes two particles with the same mass living in one spatial dimension which interact with each other by means of a force depending only on the mutual distance. If we call Ux (x) ≡ dUdx(x) the standard equations can be written as m¨ q1 = −Ux (q1 − q2 ) m¨ q2 = Ux (q1 − q2 ). These equations are particularly easy to manage, since they can be easily de-coupled simply by defining two new independent variables X ± ≡ q1 ± q2 . 14

In this way we get ¨+ = 0 X ¨ − = − 2 Ux (X − ). X m

(3.5)

On the other side the non-standard equations obtained by (3.2) are 2q1,k − q1,k−1 − q1,k+1 ≈ Ux (q1,k − q2,k ) η2 2q2,k − q2,k−1 − q2,k+1 m ≈ −Ux (q1,k − q2,k ). η2 m

Again it is possible to de-couple these equations: let us define Xk± ≡ q1,k ±q2,k , for all k. We obtain + + 2Xk+ − Xk−1 − Xk+1 ≈0 2η 2 − − 2Xk− − Xk−1 − Xk+1 ≈ Ux (Xk− ), m

(3.6)

which are, of course, just the non-standard version of (3.5). As far as we do not fix the form of the potential we cannot say anything more. We are now going to consider two different choices of U , and see what happens. Example 3A: U (q1 − q2 ) = 12 (q1 − q2 )2 With this choice of potential the non trivial equation in (3.5) becomes ¨ − + ω 2 X − = 0, where ω = 2k . the equation of an harmonic oscillator X m We fix the following boundary conditions: q1 (0) = q2 (0) = 0 and q1 (1) = q2 (1) = 1. It may be worthwhile to remind that we must use a certain care to fix the boundary conditions if we want to obtain an unique solution of the variational problem (as it usually happens when we deal with harmonic oscillators). The standard equations for X + and X − are easily solved and, once the boundary conditions above are implemented, we get the unique solution q1 (t) = q2 (t) = t. 15

It may be interesting to observe that the different set of boundary conditions q1 (0) = q2 (0) = 0 and q1 ( ωπ ) = q2 ( ωπ ) = 0 would give instead the infinite set of solutions q1 (t) = −q2 (t) = D sin t, where D is an arbitrary constant. Let us now move to the non-standard approach. How can be deduced from the results contained in the previous Section, the equation in (3.6) for the variable Xk+ has the solution X + (t) = A + Bt, where A and B are arbitrary constants to be fixed by the boundary conditions. In fact, introducing the + = 0, shift operator, we can rewrite this equation as (Eη2 − 2Eη + 1)Xk−1 which coincides with equation (2.7) with a = b = 0. The equation for Xk− , t/η t/η − − 2Xk− − Xk−1 − Xk+1 ≈ ω 2 η 2 Xk− , is solved by X − (t) = Cs1 + Ds2 where, again, C and D are constants and √ 2 − ω 2 η 2 ± ωη ω 2 η 2 − 4 s1,2 ≡ . 2 t/η

t/η

First of all it is interesting to observe that st[s1 ] = eiωt and st[s2 ] = e−iωt , so that the general solution strongly resembles the standard one. Furthermore, once the boundary conditions q1 (0) = q2 (0) = 0 and q1 (1) = q2 (1) = 1 are fixed, and once the standard part of the complete solution is taken, we obtain back exactly the standard solution. Example 3B: U (q1 − q2 ) = −(q1 − q2 )3 {1 + (q1 − q2 )} This is a second example fitting in the class of Example 3, and again it can be exactly solved. The first equation in (3.5) gives the usual free solution X + (t) = A + Bt, ¨ − = 3(X − )2 + 4(X − )3 is a bit more difficult and can while the second one, X be solved introducing a new variable Z(X) ≡ X˙ − (t). We have put m = 2 for simplicity of notation. Fixing the initial conditions q1 (0) = q˙1 (0) = q˙2 (0) = 0 and q2 (0) = 1 we get the solutions q1 (t) = 21 + t21−2 and q2 (t) = 12 − t21−2 . Let us move to the non-standard approach. In the same unit m = 2 the system (3.6) can be rewritten as + + 2Xk+ − Xk−1 − Xk+1 ≈0 − − 2Xk− − Xk−1 − Xk+1 ≈ −η 2 (3(Xk− )2 + 4(Xk− )3 ).

16

The first equation is the usual free particle equation, whose solution is X + (t) = A + Bt, while the second is non-linear and can be solved by defining X − −X −

a new set of variables Zk ≡ k+1η k . In these new variables the equation − can be rewritten as (Zk+1 − Zk )Zk ≈ (3(Xk− )2 + 4(Xk− )3 )(Xk+1 − Xk− ), whose solution is 12 Zk2 = (Xk− )3 + (Xk− )4 , how it is easily verified. This last equation can be further managed and it gives the solution X − (t) = t22−2 . Fixing the same initial conditions as in the standard situation we obtain again the same solutions we already got.

4

The Inverse Problem: a First Approach

We devote this Section to discuss a first approach to the inverse problem of the calculus of variation. The standard inverse problem is concerned with the possibility of finding a lagrangian L which, by means of the Eulero-Lagrange equations (3.3), returns a given set of differential equations. In the nonstandard version of this problem we simply replace the set (3.3) with its non-standard counterpart (3.2). We will not consider the general situation, which is quite a difficult task to solve, but only a certain number of models for which a solution can be easily found. These models will be chosen also for their physical relevance; indeed they describe classical equations of motion for point particles. We will also obtain a no-go result. We refer to Appendix B for a brief resume of the standard approach to the inverse problem. Let us consider the following system of first order non-standard difference equations   q1,k+1 − q1,k ≈ ηφ1 (q1,k , q2,k , ..., qn,k , tk )         q2,k+1 − q2,k ≈ ηφ2 (q1,k , q2,k , ..., qn,k , tk )

..................................

    .................................     

(4.1)

qn,k+1 − qn,k ≈ ηφn (q1,k , q2,k , ..., qn,k , tk ),

which is the non-standard version of the system (B.1) in Appendix B. Here 17

η is an infinitesimal. In Appendix B we have discussed some results related to the standard approach. In this Section we will discuss the analogous of those results in the non-standard approach. First of all we recall that any n-th order difference equation can be rewritten in this form, simply by introducing new variables, exactly in the same way we do for an n-th order differential equation. We start discussing the existence of a first order lagrangian L(q1 , .., qn , q˙1 , .., q˙n , t) =

n X

q˙α fα (q1 , .., qn , t) + f0 (q1 , .., qn , t),

(4.2)

α=1

which, together with the system (3.2), returns system (4.1). The details are long and not so interesting and therefore will be omitted. Assuming that all the functions fβ are twice differentiable it is a straightforward exercise to obtain the non-standard version of equation (B.4). Defining fα (q1,k , .., qβ,k+1 , .., qn,k , tk ) − fα (q1,k , .., qβ,k , .., qn,k , tk ) − qβ,k+1 − qβ,k fβ (q1,k , .., qα,k + τ, .., qn,k , tk ) − fβ (q1,k , .., qα,k , .., qn,k , tk ) − (4.3) τ

k Mα,β ≡

and fα (q1,k , .., qn,k , tk+1 ) − fα (q1,k , .., qn,k , tk ) + η f0 (q1,k , ., qα,k + τ, ., qn,k , tk ) − f0 (q1,k , .., qα,k , .., qn,k , tk ) + , (4.4) τ

Fαk (q1,k , ., qα,k , .., qn,k , tk , τ, η) ≡ −

then the functions fβ in the lagrangian must be taken to be any set of solutions of the system n X

k φβ = Fαk Mαβ

α = 1, 2, .., n,

k = 0, 1, 2, .., ∆ − 1.

(4.5)

β=1

Before considering some examples in which the lagrangian can be found, we state the announced no-go result. If n is an odd integer, then it is easy k , is to see that the skew symmetric matrix M k , with matrix elements Mαβ 18

singular for any k. It is therefore not possible to invert system (4.5) in order to return to the original system (4.1). The situation changes drastically if n is an even integer, since a skew symmetric matrix with an even number of rows and columns may have, in general, a non zero determinant. This result is in agreement with its known standard version: in both cases, for odd n, there exists no first-order lagrangian solving the inverse problem. Now we analyze in some details the non-standard version of the results discussed in Appendix B for the easiest non trivial situation: n = 2. This is not an useless task since, as it is well known, (almost) any equation of motion of a single classical particle can be rewritten in this form. The system we will deal with is therefore   q1,k+1 −q1,k ≈ φ (q , q ) 1 1,k 2,k η  q2,k+1 −q2,k ≈ φ2 (q1,k , q2,k ) η

(4.6)

where the functions on the right hand sides are now supposed to be time independent, and the tentative lagrangian is therefore L(q1 , q2 , q˙1 , q˙2 ) = q˙1 f1 (q1 , q2 ) + q˙2 f2 (q1 , q2 ) + f0 (q1 , q2 ).

(4.7)

Let us notice that, due to time independence of the φα , even the first order lagrangian can be searched with no explicit dependence on t. The system (4.5) reduces now to the two equations   M k φ (q , q ) ≈ F k 1,2 2 1,k 2,k 1  M k φ1 (q1,k , q2,k ) ≈ F k 2,1

(4.8)

2

k k and Fαk are defined in (4.4) but do not depend explicitly ≈ −M2,1 where M1,2 on t. We start now considering different classes fixed by particular choices of the functions φα , the same discussed in Appendix B from a standard point of view. We will show that in these situations system (4.8) can be solved and the functions fα give rise to the same lagrangian as in the standard situation.

19

Condition 1: φ1 (q1 , q2 ) = φ1 (q2 ) φ2 (q1 , q2 ) = φ2 (q1 ) k If we put f1 (q1 , q2 ) = q2 and f2 (q1 , q2 ) = 0 from the definition of M1,2 it k k follows that M1,2 = 1 = −M2,1 , so that the system (4.8) take the form   1 φ (q ) ≈ F k 2 1,k 1  −1 φ1 (q2,k ) ≈ F k . 2 R

From the definition of Fαk it immediately follows that f0 (q1 , q2 ) = φ2 (q1 ) dq1 − R φ1 (q2 ) dq2 , so that the lagrangian coincides with the one given in Appendix B. A simple example of this situation is given by the following system:   q1,k+1 −q1,k ≈ q 2 2,k η  q2,k+1 −q2,k ≈ q1,k + 1. η

Due to the results above, it is immediate to find the lagrangian for this set of non-standard equations: L(q1 , q2 , q˙1 , q˙2 ) = q˙1 q2 + 21 q12 + q1 − 13 q23 . We can verify the rightness of this result simply by substituting this lagrangian in equation (3.2). Condition 2: φ1 (q1 , q2 ) = q2 φ2 (q1 , q2 ) = φ2 (q1 ) This is actually a particular situation of the previous case. We prefer to consider this situation separately since it describes the well known standard equation of motion x¨ = f (x). Of course the solution for the functions fβ directly follows from the results above: f1 (q1 , q2 ) = q2 , f2 (q1 , q2 ) = 0 and R q2 f0 (q1 , q2 ) = φ2 (q1 ) dq1 − 22 . Let us now consider a simple example. We consider the following system   q1,k+1 −q1,k ≈ q 2,k η  q2,k+1 −q2,k ≈ −q1,k . η

This is the non-standard version of the well known differential equation x¨ = −x. Using the above definition we find the expression of our Lagrangian: 20

L(q1 , q2 , q˙1 , q˙2 ) = q˙1 q2 − 12 (q12 + q22 ). Again, if we use this lagrangian in (3.2) we obtain back the system above. Condition 3: φ1 (q1 , q2 ) = q2 φ2 (q1 , q2 ) = φ2 (q2 ) This is a classical equation of motion of the following type: x¨ = f (x), ˙ therefore it describes a one-dimensional particle subjected to a given friction. R Again we put f2 (q1 , q2 ) = 0, and f1 (q1 , q2 ) = − φ2dq(q22 ) . In this way the system (4.8) take the form   

−1 φ2 (q2,k ) 1 φ2 (q2,k )

φ2 (q2,k ) ≈ F1k ≈ F2k .

q2,k

R

which is solved by f0 (q1 , q2 ) = −q1 + φq22 (qdq22) . An example of this situation is given by   q1,k+1 −q1,k ≈ q 2,k η  q2,k+1 −q2,k ≈ q 2 . 2,k

η

Using the results above we deduce the following form for the lagrangian: L(q1 , q2 , q˙1 , q˙2 ) = qq˙21 − q1 + log |q2 |. Again, it is a simple exercise to obtain the system above from this lagrangian by means of equations (3.2). Condition 4: φ1 (q1 , q2 ) = q2 φ2 (q1 , q2 ) = Φ(q1 ) Ψ(q2 ) This situation describes a classical equation of motion of the following type: x¨ = Φ(x) Ψ(x). ˙ This time it is convenient to put f1 (q1 , q2 ) = 0 and q1 f2 (q1 , q2 ) = Ψ(q2 ) . In this way the system (4.8) becomes   

−1 Ψ(q2,k ) 1 Ψ(q2,k )

Φ(q1,k ) Ψ(q2,k ) ≈ F1k ≈ F2k .

q2,k R

R

2 dq2 which is solved by f0 (q1 , q2 ) = − Φ(q1 ) dq1 + qΨ(q . 2) An example is given by the following system of non-standard equations

  q1,k+1 −q1,k ≈ q 2,k η  q2,k+1 −q2,k ≈ q 2 q2,k . 1,k η

21

In this example the definitions above return L(q1 , q2 , q˙1 , q˙2 ) =

q˙2 q q2 1

+ q2 −

q13 . 3

Condition 5: φ1 (q1 , q2 ) = φ1 (q1 ) φ2 (q1 , q2 ) = φ2 (q2 ) This situation does not describe a second order equation of motion but rather two disaccoppied equations q˙1 = φ1 (q1 ) and q˙2 = φ2 (q2 ). Due to the no-go result discussed in this Section we know that it is not possible to find a first order lagrangian returning this set of equations as a sum of two contributions, each depending on only one variable. Nevertheless a solution can be found when q1 and q2 are considered, in a sense, as accoppied variables. R dq2 1 and f2 (q1 , q2 ) = 0. The system (4.8) can Let us put f1 (q1 , q2 ) = φ1 (q ) φ2 (q2 ) 1 be now rewritten as   

1 φ1 (q1,k )φ2 (q2,k ) −1 φ1 (q1,k )φ2 (q2,k )

φ2 (q2,k ) ≈ F1k φ1 (q1,k ) ≈ F2k .

which is immediately seen to be solved by f0 (q1 , q2 ) = An example is given by the system

R

dq1 φ1 (q1 )

−

R

dq2 . φ2 (q2 )

  q1,k+1 −q1,k ≈ q1,k + 1 η  q2,k+1 −q2,k ≈ 1 . η

q2,k

The lagrangian is now L(q1 , q2 , q˙1 , q˙2 ) =

q˙1 q22 2(q1 +1)

+ log |q1 + 1| −

q22 . 2

It is worthwhile to remind that all the integrals above, which are assumed to exist, must be understood in the non-standard sense. We have considered here five class of difference equations for which it is simple to find a first order lagrangian. Different classes can also be treated: we will consider these generalizations in a future paper. We also wish to use NSA to discuss the variational principle in quantum mechanics: δ < Ψ, HΨ >= 0,

22

where Ψ is the wave function and H is the hamiltonian of the system. We expect to obtain, with some appropriate procedure, the non-standard analogous of the Schroedinger equation. Further generalizations will hopefully produce also the non-standard relativistic (Dirac or Klein-Gordon) equations of motion.

Acknowledgments It is a pleasure to thank Prof. S. Valenti, to whom also belongs the idea of approaching classical mechanics with non-standard techniques, for stimulating discussions. Thanks are also due to Dr. R. Belledonne for her kind reading of the manuscript.

23

A

Appendix: Solutions of Some Standard Difference Equations

In this Appendix, included here for a better readability of this paper, we will briefly summarize some techniques of solution of few particular difference equations. We refer to [8] for a simple but detailed analysis. Let E be the shift operator acting on regular functions in the canonical way: Eu(x) ≡ u(x + 1). For a general homogeneous linear equation of order n with real constant coefficients of the form L(E)u(x) ≡ (b0 E n + b1 E n−1 + ... + bn )u(x) = 0

(A.1)

the solution must be searched of the form u(x) = Cemx , where C is a constant and m is fixed by the equation. Calling q ≡ em the equation becomes b0 q n + b1 q n−1 + ... + bn = 0, which gives three different situations, depending on the solutions qα 0 s of the equation: (i) the n roots are all different. In this case the general solution of equation (A.1) is u(x) = C1 q1x + C2 q2x + .... + Cn qnx , where the Cn are arbitrary constant. (ii) Let us suppose that one root, q1 , is complex. Therefore, if the coefficients bα are real, also q2 ≡ q1 is a solution of the equation. The general solution of (A.1) is now u(x) = C1 q1x + C2 q1 x + .... + Cn qnx (iii) The third situation appears, for instance, when all the roots but two are real and distinct while q1 = q2 . In this case the general solution is u(x) = C1 q1x + C2 xq1x + .... + Cn qnx . If the equation is not homogeneous we have to add to the general solution of (A.1) a particular solution of the complete equation. As we can see the theory (and the practice) of the difference equations strongly resembles the theory of the ordinary differential equations. 24

A particular solution of the complete equation, L(E)u(x) = Φ(x),

(A.2)

can be very easily found for particular examples of the function Φ(x). For 1 instance if Φ(x) = ax then a particular solution is up (x) = L(a) ax , whenever L(a) 6= 0. Of course, due to the linearity of the operator L, it is very easy to find a particular solution even if we have Φ(x) = cos(αx) or Φ(x) = sin(αx). Moreover, following the common sense, when Φ(x) is a polynomial of degree m, Pm (x), we look for a particular solution of (A.2) as a polynomial of the same degree, Qm (x). Still, if Φ(x) = β x Pm (x), then we have up (x) = β x Qm (x). And yet, for Φ(x) = β x sin(αx) or Φ(x) = β x cos(αx) we search for a solution of the form up (x) = β x (A cos(αx) + B sin(αx)). More information, techniques and tricks can be found in [8] and in many more textbooks of numerical analysis.

B

Standard Inverse Calculus

This Appendix is devoted to remind to the reader some information about the standard inverse calculus of variation, information which have been translated in Section 4 in our non-standard language. We do not plain to give a general overview on this subject, widely discussed in the literature, see, for instance, [9, 10, 11] and references therein. Let us consider the following system of first order differential equations:   x˙ 1 = φ1 (x1 , x2 , ..., xn , t)         x˙ 2 = φ2 (x1 , x2 , ..., xn , t)

.. .............................

    .. .............................     

(B.1)

x˙ n = φn (x1 , x2 , ..., xn , t)

As it is well known any differential equation of order n can be rewritten in this form by simply introducing new variables. It is discussed in the literature 25

that not all the systems like (B.1) can be obtained by a quadratic lagrangian, see [11]. Let us therefore consider a first order lagrangian for such a system, L(x1 , .., xn , x˙ 1 , .., x˙ n , t) =

n X

x˙ α fα (x1 , .., xn , t) + f0 (x1 , .., xn , t),

(B.2)

α=1

where all the functions fβ do not depend on x˙ j . Using the standard EuleroLagrange equations (3.3) we obtain a certain number of constraints for the functions fβ . These are necessary conditions for the lagrangian in (B.2) to give back the set (B.1). Defining Mαβ ≡

∂fβ ∂fα − ∂xβ ∂xα

Fα ≡

∂f0 ∂fα − ∂xα ∂t

(B.3)

it is easy to see that the following set of n equations must be satisfied: n X

Mαβ φβ = Fα

α = 1, 2, .., n.

(B.4)

β=1

A first interesting consequence of this constraint is that, if n is an odd integer, then it does not exist any first order lagrangian for the system (B.1). The reason is simply that the skew-symmetric matrix M is necessarily singular if the number of the rows is odd. The situation is different if n is an even integer, since in this case in general we have det M 6= 0. In [9] it is discussed that, once f0 is fixed it surely exists the solution fα of the system (B.4). In this paper we have focused our attention on second order differential equations since they have a particular relevance in classical mechanics. We consider therefore the following particular form of the system (B.1):   x˙ = φ (x , x , t) 1 1 1 2  x˙ 2 = φ2 (x1 , x2 , t)

(B.5)

We list here some particular situations in which the functions f0 , f1 , f2 can be easily found. It is very simple to verify this result and we will not do it here. 26

We assume that all the quantities below are assumed to be well defined. 1)

φ1 (x1 , x2 , t) = φ1 (x2 ) φ2 (x1 , x2 , t) = φ2 (x1 ) ⇒ Z

L(x1 , x2 , x˙ 1 , x˙ 2 , t) = x˙ 1 x2 +

Z

φ2 (x1 ) dx1 −

φ1 (x2 ) dx2

2)

φ1 (x1 , x2 , t) = φ1 (x1 ) φ2 (x1 , x2 , t) = φ2 (x2 ) ⇒ Z Z 1 Z dx2 dx1 dx2 L(x1 , x2 , x˙ 1 , x˙ 2 , t) = x˙ 1 + − φ1 (x1 ) φ2 (x2 ) φ1 (x1 ) φ2 (x2 ) 3) φ1 (x1 , x2 , t) = x2 φ2 (x1 , x2 , t) = φ2 (x1 ) ⇒ Z x2 L(x1 , x2 , x˙ 1 , x˙ 2 , t) = x˙ 1 x2 + φ2 (x1 ) dx1 − 2 2 4) φ1 (x1 , x2 , t) = x2 φ2 (x1 , x2 , t) = φ2 (x2 ) ⇒ Z Z dx2 x2 dx2 L(x1 , x2 , x˙ 1 , x˙ 2 , t) = x˙ 1 + − x1 φ2 (x2 ) φ2 (x2 ) 5) φ1 (x1 , x2 , t) = x2 φ2 (x1 , x2 , t) = Φ(x1 )Ψ(x2 ) ⇒ Z Z x2 dx2 x˙ 2 x1 L(x1 , x2 , x˙ 1 , x˙ 2 , t) = − Φ(x1 ) dx1 + Ψ(x2 ) Ψ(x2 ) Some brief remarks before concluding: (i) Situation 2) describes two decoupled equations; (ii) Situations 3), 4) and 5) describe three different examples of classical equations of motion, so that they have a direct physical interpretation; (iii) whenever the functions φα do not depend explicitly on t, even the functions fα may be chosen to be time-independent.

27

References [1] F. Bagarello, Non-Standard Variational Calculus with Applications to Classical Mechanics 1: an Existence Criterion, submitted to Il Nuovo Cimento B [2] A. Robinson, Non Standard Analysis, North-Holland Publ. Co., (1966). [3] A. Robert, NonStandard Analysis, John Wiley and Sons, New York (1985). [4] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York (1962). [5] E. Isaacson, H.B. Keller, Analysis of Numerical Methods, John Wiley and Sons, New York,(1966). [6] G.M. Phillips, P.J. Taylor, Theory and Applications of Numerical Analysis, Academic Press: London and New York (1973). [7] I.M. Gelfand, S.V. Fomin, Calculus of Variations, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1963). [8] L.A. Pipes, L.R. Harvill Applied Mathematics for Engineers and Physicists, McGraw-Hill, New York, (1970). [9] S. Hojman, L. F. Urrutia, J. Math. Phys., 22, (1981), 1896-1903. [10] M. Henneaux, Ann. Phys., 140, (1982), 45-64. [11] S. Hojman, F. Pardo, L. Aulestia, F. de Lisa, J. Math. Phys., 33, (1992), 584-590.

28