Inverse problem in Lagrangian dynamics: special solutions for potentials possessing families of regular orbits on a given surface

Abstract

We study the following version of the inverse problem in Lagrangian dynamics: Given a mono-parametric family of regular curves f(u,v)=c, with a “slope function” γ=fv/fu, on a smooth surface S: (---671----1) , we determine all the potential functions V=F(γ) which possess these curves as trajectories. We find a necessary and sufficient condition which must be satisfied for the “slope function” γ so as the problem has a solution. We examine many cases of orbits on different surfaces and with the use of this condition, we ascertain that the problem may have a solution or not, depending on the given surface S and the corresponding mono-parametric family of regular curves lying on it. Several examples are worked out and pairs (γ, F(γ)) are found. Special cases are examined too.

Keywords:

• Inverse problem of Lagrangian dynamics
• Mono-parametric families of orbits
• Surfaces
• Two-dimensional potentials
• Partial differential equations
• 2000 Mathematics Subject Classifications:
• 70F17
• 70M20
• 53A05
• 35F05
• 35G05
• 35G20

1. Introduction

Szebehely [1] published a partial differential equation for the potential function V = V(x, y) which produces a mono-parametric family of planar orbits f(x, y) = c, and the energy E of them is given in advance as a function of the constant c, namely E = E(c). This equation was studied then by several authors (e.g. Bozis [2], Puel [3]). Bozis [4] presented a second-order linear partial differential equation giving the potential functions V=V(x,y) which give rise to a preassigned family of planar curves f(x, y) = c. Bozis' equation does not include the energy E and consequently no assumption about the energy dependence E=E(f) needs to be made.

Mertens [5] extended Szebehely's method to a family of curves f(u, v) = c on a surface S in 3D space and obtained a linear partial differential equation in the potential function V(u, v). Moreover, Bozis and Mertens [6] derived a second-order partial differential equation of hyperbolic type in the unknown potential function V=V(u,v) in which all the coefficients are known functions of the curvilinear coordinates (u, v) and gave some examples. A generalization of Szebehely's problem to account for holonomic conservative mechanical systems with n-degrees of freedom was made by Melis and Borghero [7] and by Borghero and Melis [8]. A review of the basic facts of inverse problem in dynamics was made by Bozis [9]. Solvable cases of the planar inverse problem were presented in Grigoriadou et al. [10]. The determination of potentials having two families of orthogonal trajectories with the use of Joukovsky's formula was studied by Puel [11].

Borghero and Bozis [12] studied the mono-parametric isoenergetic families of planar orbits f(x,y)=c created by homogeneous potentials V(x, y). Kotoulas [13] found solvable cases of the PDE in V(u, v) given by Bozis and Mertens [6]. Moreover, Kotoulas [14] determined the generalized force field which gives rise to a two-parametric family of orbits on a given surface. Several papers of the last decade may be found in Anisiu's report [15]. Anisiu [16] derived in a unified manner the two basic equations of the inverse problem in dynamics (for more details see Bozis [9]). Recently, Bozis and Anisiu [17] dealt with a solvable version of the planar inverse problem and found potentials of special type V = V(γ), γ=fy/fx.

In the present work we address the following question: Given a mono-parametric family of regular curves f(u,v)=c (with a “slope function” γ=fv/fu) on a smooth surface S: (---671----2) , is there any potential V = F(γ(u,v)) producing this family of curves as trajectories? Here, we have to clear out that the trajectories are bound to a given surface by external constraints. The answer to the above question depends on the regular surface S and the mono-parametric family of regular curves which lies on it. To our knowledge there are not enough results on this version of inverse problem in the literature.

In section 2 we give a full description of this problem; we find a new, ordinary differential equation of second order (equation (10)(---671--10) ) which the function F=F(γ) of its unique argument must satisfy and then we proceed to a necessary and sufficient condition, i.e. condition (12)(---671--12) , which must be fulfilled for the “slope function” γ=fv/fu in order the problem admits of a solution. In section 3 we classify solvable cases and give pertinent examples. In the generic case it is shown that if a solution V=F(γ) exists, then it is determined uniquely, up to an additive and multiplicative factor. In section 4 we study several special cases which arise from the generic case. In section 5 we make a substantial effort to find more general solutions considering special families γ=γ(u,v) on specific surfaces. Finally, the conclusions and some implications are presented in section 6.

2. A differential condition for the “slope function” γ

In an Euclidean 3D-space E³ with an orthonormal Cartesian system of reference Oxyz we assign a smooth surface S: (---671--1) with u, v as curvilinear coordinates on S. On this surface, we consider a mono-parametric family of regular curves given in the solved form (---671--2) where c = const ≠ 0 is the parameter of the family (2).

For the given family of orbits we define γ as follows: γ=fv/fu. The “slope function” γ represents the family (2)(---671--2) in the sense that if the family (2)(---671--2) is given, then γ is determined uniquely. On the other hand, if γ is given, we can obtain a unique family (2). The inverse problem of dynamics consists in finding potentials V which can give rise to this family of orbits (2)(---671--2) on a given surface (1)(---671--1) (for the theory and application of inverse and ill-posed problems, see Yagola et al. [18]).

The line-element on the surface S in this system of parameters is given by: (---671--3) where g11,g12,g22 are the components of the metric tensor.

Now, we consider a particle of unit mass which describes any member of the given family (2). The kinetic energy (T) of the test particle is given by (---671--4) where the dot denotes differentiation with respect to time.

Mertens [5] derived a linear, first-order partial differential equation for the potential function V=V(u,v) for any preassigned dependence E=E(f), of the total energy E of the given family f=f(u,v). This equation is the following one: (---671--5) where W is given in the Appendix.

Moreover, Bozis and Mertens [6] produced a linear, second-order partial differential equation in V=V(u,v) which is independent of the total energy E and gives all the potential functions generating family (2)(---671--2) on the given surface (1). The total energy must be constant along each orbit; so E = E (f). Thus, we have: Ev = Effv and Eu = Effu. Assuming W≠0 and with the use of the fact that Ev=γEu, Bozis and Mertens [6] obtained the following equation: (---671--6) and the coefficients kj,β(j=1,…,4) are given in the Appendix. The subscripts denote partial differentiation with respect to the corresponding variable. This is a second-order partial differential equation in the unknown function V = V(u, v) of hyperbolic type. A classification of PDEs of second order can be found in [19, Chapter 3]. From (6)(---671--6) it is easy to check that if V is a solution, then V′ = c1V +c2 is a solution too (c1,c2 are constants). So, without loss of generality, we shall omit these constants below.

In the present work we shall reconsider the previous equation (6)(---671--6) and work with it. We have to solve of a linear, second-order PDE in V(u, v) and we look for triplets {γ,(gij),V(u,v)} (i,j=1,2) which satisfy it. Especially, we are interested in solutions of the type V=F(γ(u,v)). This type of solution makes the mathematical calculations simpler. Solutions of the form V = V(γ) (γ = fy/fx) for the planar inverse problem of dynamics were recently found by Bozis and Aniciu [17].

Firstly, we compute the partial derivatives of first and second order of the potential function V=V(u,v) and then we substitute them into (6)(---671--6) . We have: (---671--7) Then, equation (6)(---671--6) reads: (---671--8) where (---671--9) Supposing that M2≠0, equation (8)(---671--8) becomes: (---671--10)

Since F = F(γ), the ratio M=M1/M2 in (10)(---671--10) must depend only on the “slope function” γ. Hence, by necessity, we have: M=m(γ(u,v)) and consequently (---671--11) Equation (11)(---671--11) is written equivalently as follows: (---671--12) where (---671--13)

Equation (12)(---671--12) is a new, nonlinear partial differential equation of third order in the unknown function γ = γ(u, v) and it is very difficult to find all the solutions of it. This equation also includes the components of the metric tensor which are different in any considered surface. We shall set here: τ=γvMu−γuMv. τ=0 is equivalent to (12)(---671--12) .

Moreover, equation (12)(---671--12) is the necessary and sufficient condition for the “slope function” γ so as equation (10)(---671--10) has a solution V=F(γ). Indeed, if γ is a solution of (12)(---671--12) for which M2≠0, we can get from (10)(---671--10) (---671--14) with c1,c2 constants and V = F(γ(u,v)). We shall omit the constants c1,c2 in the following as we explained above. The meaning of the solution (14)(---671--14) is that we can find the potential function V = V(u, v) by quadratures. So, it is important for us to focus on (12)(---671--12) from which we can and must select adequate families γ=γ(u,v) on given surfaces for which a compatible potential will be found from (14)(---671--14) .

3. Certain classes of families on given surfaces

In this section we shall deal with special forms of the “slope function” γ which can give a solution to the problem. Let us start with Case 1 γ=h(s1u+s2v+s3) where h is an arbitrary function of its argument w = s1u+s2v+s3 and s1,s2,s3=const ≠0. Thus, we have: γu = λγv with λ = s1/s2. Consequently, the necessary and sufficient condition (12)(---671--12) is written as: (---671--15) where Φ0 is an arbitrary function of its argument w = v + λu. So, if the ratio M =M1/M2 and the metric are of the form (15)(---671--15) , then a solution to the problem exists and it is found from (14)(---671--14) (Example 1).

Example 1 We assign the cylindrical surface S: (---671----3)  = {u, v,(u+v)²} and we considered the mono-parametric family of curves f=e^v(u+v)=c on it. Then we have: (---671--16) So, from (14)(---671--14) we find the potential function V=F(γ). It is: (---671--17) Remark 1 If we consider the mono-parametric families of curves f=e^u(u+v)=c on the same surface, then γ = 1/(u + v + 1) and τ = 0. So, the problem has a solution and this is given by: (---671--18) We note here that the case of (---671----4) is studied analogously.

Counterexample 1 We assign the surface S: (---671----5)  = {u, v,(u+v)²} and we consider the mono-parametric family of curves f=e^v(u+2v)=c on it. Then we have: (---671--19) So, no such solution to this problem exists.

Case 2 γ=h(u/v) with h an arbitrary function of its argument w=u/v. Then, we compute the partial derivatives of γ with respect to u, v and we get: γu/γv=−v/u. The necessary and sufficient condition (12)(---671--12) is written as: (---671--20) where Φ0 is an arbitrary function of its argument w. So, if the ratio M=M1/M2 and the components of the metric tensor are of the form (20)(---671--20) , then a solution to the problem exists and it is found from (14)(---671--14) (Example 2).

Example 2 We assign the circular cylinder S: (---671----6)  = {cos u, sin u,v} and we consider the mono-parametric family of curves f=(u+v)/(uv)=c on it. Then we have: (---671--21) So, from (14)(---671--14) we find the potential function V=F(γ). It is: (---671--22) Case 3 γ=h(uv) where h is an arbitrary function of its argument w = uv. Then, we compute the partial derivatives of γ with respect to u, v and we get: γu/γv=v/u. The necessary and sufficient condition (13)(---671--13) is written as: (---671--23) where Φ0 is an arbitrary function of its argument. So, if the ratio M=M1/M2 and the components of the metric tensor are of the form (23)(---671--23) , then a solution to the problem exists and it is found from (14)(---671--14) (Example 3).

Example 3 We consider the metric: ds²=du²+uvdv² (u,v>0) and the mono-parametric family of curves f=logu+v²=c on it. Then we have: (---671--24) So, from (14)(---671--14) we find the potential function V=F(γ). It is: (---671--25)

Case 4 γ=γ(v). In this case we have γu = 0 and consequently γuu = 0 and γuv = 0. Then the relations of M1 and M2 in (9)(---671--9) take the form: (---671--26) The condition (12)(---671--12) reads: (---671--27) Moreover, if we substitute (26)(---671--26) into (27)(---671--27) , we get: (---671--28) Since γv≠0, we take from (28)(---671--28) (---671--29) Hence, if for the given family (2)(---671--2) on the smooth surface (1)(---671--1) the condition (29)(---671--29) is satisfied, then a solution to the problem exists (Example 4).

Example 4 We assign the sphere (---671----7) (R > 0) and the mono-parametric family of curves f=u+cos v=c on it. Then we have: (---671--30) So, from (14)(---671--14) we find the potential function V=F(γ). It is: (---671--31) We note here that we have examined many other cases of curves on sphere and we have found solutions V=F(γ). These families of curves are: f(u,v)=u+G(v)=c with G(v)=cosv, tanv,secv,e^v, v².

Remark 2 We note here that the case γ=γ(u) is studied in an analogous way. More general solutions are given in section 5.

4. Special cases

Here, we shall study some special cases which arise from the fact that some quantities were taken as non-zero up to now. We shall begin with

Case A M2=0. Another expression for M2 in (9)(---671--9) is: M2=(αγu+βγv)(γγu−γv). So, M2 equals to zero if and only if

1.	A1: αγu +βγv=0 or
2.	A2: γγu−γv =0

In the subcase A1 we differentiate the relation αγu+βγv = 0 twice with respect to u and v respectively, we find γuu,γuv and then we replace them into the expression of M1 in (10)(---671--10) . We have: (---671--32) Then, after some straightforward algebra, we obtain for M1 the following simple form: (---671--33) This means that M1 is always non-zero. So, from (9)(---671--9) we take the trivial solution (---671--34) In the subcase A2, namely, if we have a “slope function” γ such that γγu − γv = 0, then M2 = 0 and M1 ≠ 0. This is similar to the previous subcase A1 and the solution F(γ) = c2 = const is always found.

Case B W = 0. If for the given family of orbits (2)(---671--2) on the considered surface (1)(---671--1) is W = 0, then the coefficients α and β appeared in (7)(---671--7) (given also in the Appendix) are not defined. This leads to the study of geodesic lines produced by certain potentials (Kotoulas [20]).

5. Some more general solutions to the problem

Here, we shall seek for more general solutions to the problem considering specific surfaces.

Case I We assign the helical surface (---671----8)  = {ucosv,u sinv,v} and the mono-parametric family of curves f(u,v)=H(u)+v=c on it (H is an arbitrary C³-function). Then we have (---671--35) where K is an expression of u,H′(u),H”(u),H”′(u) (not given here). The primes denote the derivatives of j-order (j = 1, 2, 3) of the arbitrary function H with respect to its argument u. So, according to (35)(---671--35) , there always exists a solution to the problem and it is found from (14)(---671--14) .

Case II We assign the surface (---671----9)  = {u, v, u^p v^q}(p,q=const ≠0) and the mono-parametric families of curves f(u,v)=u^p v^q=c on them. Then we have (---671--36) So, for any values of p, q we can find a solution to the problem. From (14)(---671--14) we get: (---671--37)

6. Conclusions

In the present study we dealt with a version of the inverse problems in Lagrangian dynamics considering regular orbits f(u,v)=c(c≠0) on smooth surfaces. We sought for solutions of the type V=F(γ). The main results of our work are summarized below:

In section 2 we found a necessary and sufficient condition (12)(---671--12) which must be satisfied by the “slope function” γ. This condition is the basic result of our study. Moreover, in section 3, we examined several cases for which solutions do exist and are found from (14)(---671--14) . In the present work we extended the study of inverse problem considering regular orbits on a given surface. An important piece of evidence in our study was the components of the metric tensor which are different in any surface. So, we may have the solution to the problem or not (Example 1 – Counterexample 1). Special cases were also studied (section 4) but we did not find any interesting solutions. The only acceptable ones were F(γ) = c3 = const.

In order to find more and important solutions of the basic equation (12)(---671--12) , we considered specific surfaces and took mono-parametric families of special type on them. As we saw in section 5, several pairs (γ,F(γ)) arise. All the computations were aided by MATHEMATICA 5.1.

The implications of this problem are mostly of a mathematical nature. Indeed, only for adequate “slope functions” γ and for adequate metrics which satisfy the condition (12)(---671--12) , equation (6)(---671--6) has solutions of the form V=F(γ(u,v)). From the physical view point, the meaning of this assumption is that for γ(u,v)=const it is also V(u, v) = const, namely, the isoclinic curves of the orbits on the given surface coincide with the equipotential curves (for more details, see Bozis and Anisiu [17]).

Acknowledgements

The author is grateful to the three anonymous reviewers for their good remarks and would also like to express thanks to Prof. G. Bozis (Emeritus Prof. in Dept. of Physics, University of Thessaloniki) for many useful comments and to S. Stamatakis (Associated Prof. in Dept. of Mathematics, University of Thessaloniki) for the discussions about surfaces. This work was financially supported by the scientific program “EPEAEK II, PYTHAGORAS”, No. 21878 of the Greek Ministry of Education and E.U.

Appendix A: General case of section 2

(---671---1)