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Running Headline: HARMONIC BALANCE AND JERK EQUATIONS Total Number of Pages: 30 including: 4 Tables over 5 pages.2 SummaryThe method of harmonic balance is applied to nonlinear jerk equations, which involve the third-order time-derivative. For many types of cubic nonlinearities, the method yields good estimates of the period and displacement amplitude of oscillations for a range of values of initial velocity amplitude when compared with numerical solutions. Some limitations, notably the restriction to zero initial acceleration, as well as implications and possible extensions are discussed.3

A density matrix treatment applicable to molecular electronic systems is formulated and used to derive an expression for the spin–lattice relaxation rate (T−11) of ligand nuclei in paramagnetic transition-metal complexes. In this way the unpaired spin density is introduced into the nuclear relaxation expression. The two-center dipolar integrals which occur in the expression for T1 are evaluated by a method based on solid spherical harmonic expansions whereby the Hamiltonian operator is reexpressed in terms of coordinates with respect to the orbital’s origin. The theory is used to treat 13C and 1H spin–lattice relaxation in ruthenium acetylacetonate. Good agreement is found between the calculated and experimental ratio TC1/TH1 whereas the Solomon–Bloembergen point-dipole approach yields results significantly in variance with the experimental. The theory is used to treat proton spin relaxation in a representative low-spin Fe(III) complex to demonstrate under what conditions the point-dipole approximation is invalidated.

The method of harmonic balance (HB) is employed to estimate the attributes of limit cycles of some nonlinear third-order (jerk) differential equations which are parity-invariant but not time-reversal-invariant. Two examples with cubic nonlinearities show that the HB method can give good values for the frequency and both the velocity and displacement amplitudes of a period one limit cycle.

Isospectral systems are those that have exactly the same free-vibration frequency spectrum with respect to a given boundary condition configuration. In this paper, seven different classes of inhomogeneous Euler-Bernoulli beams with continuous density and flexural rigidity functions are found that are analytically solvable and that are isospectral with a homogeneous beam in the clamped-clamped configuration. Of these exact solutions, one class is isospectral with the homogeneous case in all ten distinct configurations obtainable from clamped, sliding and two antiresonant end conditions, whereas another class is isospectral in the six configurations with clamped, hinged or free end conditions. A connection with inverse problems is discussed.

Various geometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with appropriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases of Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two-and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poisson summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.

Conservative oscillator equations which have quadratic non-linearities in both velocity and displacement and which possess an exact harmonic solution are investigated. The conserved quantity is constructed, and its zero value corresponds to the harmonic solution. The further signi"cance of the harmonic solution as corresponding to a bifurcation is revealed.

The appearances of jerk, the third time derivative of displacement, in several papers of mathematical and physical content are reviewed. Two simple nonlinear jerk functions, all of whose solutions are periodic, are introduced and solved. The first has period dependent on initial acceleration but independent of amplitude, and is related to the second-order cubic oscillator. The second, with intrinsically third-order dynamics, has period dependent also on initial displacement. A possible bearing on chaotic jerk functions is mooted.