P. G. L. Leach

Symmetry Lie algebras of nth order ordinary differential equations

Abstract We show that an nth (n ⩾ 3) order linear ordinary differential equation has exactly one of n + 1, n + 2, or n + 4 (the maximum) point symmetries. The Lie algebras corresponding to the respective numbers of point symmetries are obtained. Then it is shown that a necessary and sufficient conditon for an nth (n ⩾ 3) order equation to be linearizable via a point transformation is that it must admit the n dimensional Abelian algebra nA1 = A1 ⊕ A1 ⊕ … ⊕ A1. We discuss in detail the symmetry realizations of (n − 1)A1 ⊕s A1. Finally, we prove that an nth (n ⩾ 3) order equation q(n) = H(t, q, …, qn − 1) cannot admit exactly an n + 3 dimensional algebra of point symmetries which is a subalgebra of nA1 ⊕, gl(2, R ).

A direct approach to finding exact invariants for one‐dimensional time‐dependent classical Hamiltonians

For a classical Hamiltonian H=(1/2) p2+V(q,t) with an arbitrary time‐dependent potential V(q,t), exact invariants that can be expressed as series in positive powers of  p, I(q,p,t)=∑∞n=0pnfn(q,t), are examined. The method is based on direct use of the equation dI/dt=∂I/∂t +[I,H] =0. A recursion relation for the coefficients fn(q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in p are determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.

First integrals for the modified Emden equation q̈+α(t) q̇+qn =0

It is shown that the modified Emden equation q+α(t)q+qn=0 possesses first integrals for functions α(t) other than kt−1. The function α(t) is obtained explicitly in the case n=3 and parametrically for other n(≠2). The case n=2 is seen to be particularly difficult to solve.

Lie algebras associated with scalar second‐order ordinary differential equations

Second‐order ordinary differential equations are classified according to their Lie algebra of point symmetries. The existence of these symmetries provides a way to solve the equations or to transform them to simpler forms. Canonical forms of generators for equations with three‐point symmetries are established. It is further shown that an equation cannot have exactly r ∈{4,5,6,7} point symmetries. Representative(s) of equivalence class(es) of equations possessing s ∈{1,2,3,8} point symmetry generator(s) are then obtained.

Invariants and wavefunctions for some time‐dependent harmonic oscillator‐type Hamiltonians

Recently the author has shown that the Hamiltonian, H= (1/2) ωT A (t) ω+B (t)Tω+C (t), in which A (t) is a positive definite symmetric matrix and ωμ=qi, μ=1,n, i=1,n, ωμ=pi, μ=n+1,2n, i=1,n, may be transformed to the time‐independent Hamiltonian, H= (1/2) ωTω, by a time‐dependent linear canonical transformation, ω=Sω+r. H is an exact invariant of the motion described by H. A matrix invariant may also be constructed which provides a basis for the generators of the dynamical symmetry group SU(n) which may always be associated with H, usually as a noninvariance group. In this paper we examine, by way of example, an oscillator with source undergoing translation, the two‐dimensional anisotropic oscillator, general one‐ and two‐dimensional oscillators with Hamiltonians of homogeneous quadratic form and obtain explicit invariants and Schrodinger wavefunctions with the aid of the linear canonical transformations.

On the singularity analysis of ordinary differential equations invariant under time translation and rescaling

The PT is applied to the general second- and third-order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability. The effect of the two symmetries on the compatibility conditions is determined and we show that, generally, these conditions are automatically a consequence of the resonance condition. Use is made of truncated Laurent series both in ascending and descending powers. As an example, the case of the generalized Chazy equation is presented.

Generalized invariants for the time‐dependent harmonic oscillator

A generalized class of invariants, I (t), for the three‐dimensional, time‐dependent harmonic oscillator is presented in both classical and quantum mechanics. For convenience a simple notation for types of harmonic oscillator is introduced. Two interpretations, one in terms of angular momentum and the other employing a canonical transformation, are offered for I (t). An invariant symmetric tensor, Imn(t), is constructed and shown to reduce to Fradkin’s invariant tensor for time‐independent systems. The usual SU(3) (compact) or SU(2,1) (noncompact) is shown to be a noninvariance group for the time‐dependent oscillator with S{U(2) ⊗U(1) } as the invariance subgroup. Extensions to anisotropic systems and the singular quadratic perturbation problem are discussed.

On the theory of time‐dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type

Writing the canonical variables (qT, pT) as (ωT), we develop a method for transforming the time‐dependent Hamiltonian H=Aμν(t) ωμων+Bμων +C (t) to the time‐independent form H= (1/2) δμνωμων using the linear transformation ωμ=sμ ν(t) ων +rμ(t). Differential equations are obtained for the parameters sμ ν and rμ. The transformed Hamiltonian enables the construction of an invariant I and an invariant matrix [Iμν]. These invariants apply to both the classical and quantum mechanical problems. The invariant I has the dynamical symmetry group SU(n), and this characterizes all systems with Hamiltonians of the form of H.

Integration of second order ordinary differential equations not possessing Lie point symmetries

Abstract A class of integrable second order ordinary differential equations not possessing Lie point symmetries is shown to be rich in nonlocal symmetries which provide one route for integration and to have general solutions which are uniform functions.

An exact invariant for a class of time‐dependent anharmonic oscillators with cubic anharmonicity

An exact invariant is constructed for a class of time‐dependent anharmonic oscillators using the method of the Lie theory of extended groups. The presence of the anharmonic term imposes a constraint on the nature of the time dependence. For a subclass it is possible to obtain an energy‐like integral and a condition under which the motion is bounded.