Geoff Prince

A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics

Appropriate geometrical machinery for the study of time-dependent Lagrangian dynamics is developed. It is applied to the inverse problem of the calculus of variations, and a set of necessary and sufficient conditions for the existence of a Lagrangian are given, in terms of the existence of a 2-form with suitable properties, which are exactly equivalent to the Helmholtz conditions.

Dimsym and LIE: Symmetry determination packages

Dimsym and Lie are packages for the determination of various continuous symmetries of differential equations. An informal description of the programs is given and there is a discussion of the balance between algorithm and heuristic in the determination process.

Towards a geometrical understanding of Douglas's solution of the inverse problem of the calculus of variations

We describe a novel approach to the study of the inverse problem of the calculus of variations, which gives new insights into Douglass solution (1941) of the two degree of freedom case.

Symmetries of the time-dependent N-dimensional oscillator

The study of the symmetry group of the time-dependent oscillator in N dimensions with equation of motion d2xi/dt2+ Omega 2(t)xi+0, i+1, ..., N, gives the full symmetry group SL(N+2, R) of N2+4N+3 operators. The Noether subgroup consisting of 1/2(N2+3N+6) operators and the resulting constants of motion are given. A table of the commutation relations between the operators gives the structure constants of the associated Lie algebras.

Geometric aspects of reduction of order

Using the differential geometry of vectorfields and forms we reinterpret and extend the traditional idea of an integrating factor for a first order differential equation with symmetry. In particular, we provide a simple and manifestly geometric approach to reduction of order via symmetry for ordinary differential equations which largely obviates the necessity for canonical coordinates and the associated quotient manifolds. In so doing, some new results which generalise the class of Lie group actions which can be used to solve ordinary differential equations are developed

Projective differential geometry and geodesic conservation laws in general relativity. I: Projective actions

The Lagrangian structure of the geodesic equation allows an extension of classical projective geometry to one-parameter projective group actions onR×TM (whereM is the spacetime). We determine all those projective actions which arise by prolongation of oneparameter group actions onR×M. The relation between projective actions onR×TM and the equation of geodesic deviation is developed.

Adjoint symmetries for time-dependent second-order equations

The authors extend part of their previous work on autonomous second-order systems (Sarlet et al., 1987) to time-dependent differential equations. The main subject of the paper concerns the notion of adjoint symmetries: they are introduced as a particular type of 1-form, whose leading coefficients satisfy the adjoint equations of the equations determining symmetry vector fields. It is shown that all interesting properties of adjoint symmetries, known from the autonomous theory, have their counterparts in the present framework. Of particular interest is a result establishing that Lagrangian systems seem to be the only ones for which there is a natural duality between symmetries and adjoint symmetries. A number of examples illustrate how the construction of adjoint symmetries of a given system can be explored in a systematic way.

Solvable Symmetry Structures in Differential Form Applications

We investigate symmetry techniques for expressing various exterior differential forms in terms of simplified coordinate systems. In particular, we give extensions of the Lie symmetry approach to integrating Frobenius integrable distributions based on a solvable structure of symmetries and show how a solvable structure of symmetries may be used to find local coordinates for the Pfaffian problem and Darbouxs theorem.

Jacobi fields and linear connections for arbitrary second-order ODEs

Abstract We discuss generalisations of Jacobi fields and Raychaudhuri’s equation from the geodesic case to that of an arbitrary system of second-order ODEs. Our results are obtained using a natural choice of linear connection on evolution space.

A COMPLETE CLASSIFICATION OF DYNAMICAL SYMMETRIES IN CLASSICAL MECHANICS

This paper deals with the interaction between the invariance group of a second order differential equation and its variational formulation. In particular I construct equivalent Lagrangians from all such group actions, thereby successfully completing an earlier attempt of mine which dealt with some traditionally important classes of actions.