Inverse Problem Of Lagrangian Dynamics Research Articles

Bi-invariant and noninvariant metrics on Lie groups

A restricted version of the inverse problem of Lagrangian dynamics for the canonical linear connection on a Lie group is studied. Specifically for solvable Lie algebras of dimension up to and including six all algebras for which there is a compatible pseudo-Riemannian metric on the corresponding linear Lie group are found. Of the 19 such metrics four are bi-invariant. The Lie algebras are taken from tables compiled originally by Mubarakzyanov [Izv. Vyssh. Uchebn. Zaved., Mat. 4, 104–116 (1963)] and Morozov [Izv. Vyssh. Uchebn. Zaved., Mat. 4, 161–171 (1958)].

Variational connections on Lie groups

The inverse problem of Lagrangian dynamics is solved for the geodesic spray associated to the canonical symmetric linear connection on a Lie group of dimension three or less. The degree of generality is obtained in each case and concrete Lagrangians are written down.

A formula for Nijenhuis torsion and the uniqueness aspect of the inverse problem of Lagrangian dynamics

A formula is derived for the Nijenhuis tensor of an endomorphism constructed from contracting a bivector field with a 2-form. Two applications are considered: the first is to the uniqueness aspect of the inverse problem of the calculus of variations, the second is to bi-Hamiltonian systems.

Family boundary curves for holonomic systems with two degrees of freedom

The notion of the family boundary curves (FBC) is introduced for that version of the inverse problem of Lagrangian dynamics which deals with the determination of the potential V(u,v) under which a given monoparametric family of curves f(u,v)=c, on the configuration manifold (M2,g) of a conservative holonomic system with n=2 degrees of freedom, can be described as dynamical trajectories (orbits) of the representative point. It is shown that, in general, curves of the family f(u,v)=c generated by a class of potentials V(u,v) are actual orbits only in a subregion of the region where they are defined as geometrical entities. (In general, the FBC are distinct from the well known zero velocity curves (ZVC), the later referring to orbits of the same constant energy). If, however, the holonomic system is subject to non-conservative generalized forces, it is shown that we can always find many pairs of such forces (Q1,Q2) giving rise to any family of trajectories lying in any pre-assigned (open or closed) region of the configuration space. Three examples are presented to account both for conservative and non-conservative forces.

The inverse problem of Lagrangian dynamics for higher-order differential equations: a geometrical approach

Crampin and co-workers (1981, 1984) showed that, under certain circumstances, there is a class of vector fields such that if X is in this class then there is a local Lagrangian G for which X satisfies the corresponding Euler-Lagrange equations. Such an approach leads to a geometrical version of the well known inverse problem of classical mechanics. The general purpose of this work is to make a contribution problem of classical mechanics. The general purpose of this work is to make a contribution to the renewed interest in the geometric formalism of Lagrangian theories involving higher-order derivatives (generalized theories). The authors examine the above inverse problem in the framework of tangent bundle geometry of order k, k< infinity , in such a way that some of Crampin's results are re-obtained for the standard situation k=1. They extend the theory to manifolds endowed with an integrable almost tangent structure of higher order.

Second-order equation fields and the inverse problem of Lagrangian dynamics

The transformation properties of determined, autonomous systems of second-order ordinary differential equations, identified as vector fields on the tangent bundle of the space of dependent variables, are derived and studied. The inverse problem of Lagrangian dynamics is studied from this transformation viewpoint as well as the problem of alternative Lagrangians. In particular, regular Lagrangians which are analytic as functions of the first derivatives are considered. Finally, the inverse problem for second-order systems corresponding to the geodesic flow of a symmetric linear connection is investigated.

Symmetries and the inverse problem of Lagrangian dynamics for linear systems

AbstractWe discuss general, time-dependent, linear systems of second-order ordinary differential equations. A study is made of the similarities and discrepancies between the inverse problem of Lagrangian mechanics on the one hand, and the search for linear dynamical symmetries on the other hand.

The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics

Deals with the general problem of finding a multiplier matrix that can give to a prescribed system of second-order ordinary equations the structure of Euler-Lagrange equations. The approach is based on a generalisation of previous studies on linear systems. The main result concerns a set of necessary and sufficient conditions for the existence of a multiplier, which contains an infinite set of algebraic equations, the coefficients of which can be used to derive necessary conditions involving only the given right-hand sides of the differential equations. An outline is given of interesting points for future studies, and an example is presented for which all multipliers are explicitly constructed.

On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics

The conditions for a system of second-order differential equations to be derivable from a Lagrangian-the conditions of self-adjointness, in the terminology of Santilli (1978) and others-are related, in the time-independent case, to the differential geometry of the tangent bundle of configuration space. These conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second-order differential equations. Necessary and sufficient conditions for such a vector field to be derivable from a Lagrangian may be stated as the existence of a two-form with certain properties: it is interesting that it is a deduction, not an assumption, that this two-form is closed and thus defines a symplectic structure. Some other differential geometric properties of Euler-Lagrange second-order differential equations are described.

Quadratic integrals for linear nonconservative systems and their connection with the inverse problem of Lagrangian dynamics

The direct method for the construction of first integrals, developed in a previous paper, is applied to a system of second-order ordinary differential equations with time-varying coefficients. The connection between the existence of a Lagrangian and the problem of finding quadratic integrals, surmised in previous papers because of specific results, is rigorously established. Finally, the approach is illustrated by obtaining all the known quadratic integrals for a time-dependent n-dimensional harmonic oscillator from a single expression.