Finding potentials or force fields from given families of orbits is the type of inverse problem of dynamics discussed in this paper. We present the pertinent partial differential equations for various versions of the problem such as, for instance, for conservative or autonomous non-conservative fields in two or three dimensions, for inertial or relating frames, for one material point or, more, generally, for holonomic systems with n degrees of freedom. The notion of the family boundary curves is introduced. The role of the homogeneity of the given family or of the required potential, as well as the question of multitude of compatible pairs of orbits and potentials is discussed. Comments on the relation of the problem to problems of astronomical interest are made, at appropriate places, throughout the text.

The inverse problem of dynamics: basic facts

The three dimensional inverse problem for a material point of unit mass, moving in an autonomous conservative field, is solved. Given a two-parametric family of space curvesf(x, y, z)=c1,g(x, y, z)=c2, it is shown that, in general, no potentialU=U(x, y, z) exists which can give rise to this family. However, if the given functionsf(x, y, z) andg(x, y, z) satisfy certain conditions, the corresponding potentialU(x, y, z), as well as the total energyE=E(f, g) are determined uniquely, apart from a multiplicative and an additive constant.

Solution of the Three-Dimensional Inverse Problem

The second order partial differential equation which relates the potentialV(x,y) to a family of planar orbitsf(x,y)=c generated by this potential is applied for the case of homogeneousV(x,y) of any degreem. It is shown that, if the functionf(x,y) is also homogeneous, there exists, for eachm, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. Iff(x,y) is not homogeneous, in general, there is not a homogeneous potential which can create the given family; only if γ=f

 y
 /f

 x
 satisfies two conditions, a homogeneous potential does exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases.

Families of planar orbits generated by homogeneous potentials

By combining Darboux's results (1901) on the direct construction of integrable systems with results from the inverse problem of dynamics, the authors prove that if a planar potential admits a monoparametric family of conic sections with constant focal distance or a family of confocal parabolas, it is integrable and the second invariant is quadratic in the moments, while the existence of a family of straight lines, either parallel or intersecting at one point also suffices for integrability, the second invariant being linear in the momenta.

On monoparametric families of orbits sufficient for integrability of planar potentials with linear or quadratic invariants

We consider a family of curves in the n-dimensional configuration space Sn of a holonomic system with n degrees of freedom. We obtain first-order partial differential equations for the potential function U of forces under which any trajectory belonging to the given family of curves can be described by the representative point of the system. We write the potential function U supposing that, in addition to the energy integral, a first integral of motion linear in the lagrangian velocities is assigned. Next we obtain the compatibility conditions between the energy constant E, the parameter α which appears in the first integral, and the n−1 geometric constants c1, c2,⋯, cn−1 which characterize the family of trajectories. Finally we discuss two simple examples.

R. Mertens

Papers

On Szebehely's inverse problem for a particle describing orbits on a given surface