Killing Equations Research Articles

Noether's theorem for optimum control systems

Abstract In this paper Noether's theorem of classical mechanics and the variational calculus is developed for optimum control systems. Also, it is shown that the existence of first integrals of Pontryagin's maximum principle equations depend on the existence of solutions of a system of partial differential equations—generalized Killing's equations —for optimum control systems.

Symmetry in Einstein-Maxwell Space-Time

By using the complex null tetrad as basis for the tangent space, a Killing vector field (``symmetry'') is introduced into the system of Einstein's equations with Maxwell's equations. The two bivectors Fμν and Kμ;ν (the associated Killing bivector) are assumed to have a principal null direction in common. Killing's equations, Maxwell's equations, and Einstein's equations are then written down for the case where this special direction is also a principal null geodesic for the Weyl conformal tensor. A certain analog of the Goldberg-Sachs theorem is proved. The static cases, plus a sizeable class of the static algebraically special cases are examined, to wit: where the special direction is also shear-free. In particular, all such algebraically special spaces must be Petrov Type D as a result of a coupling of the principal null directions for Fμν. This algebraically special metric is derived as an example of the static classes and is a static generalization of the Reissner-Nordström metric.

Classification of linear integrals of a holonomic mechanical system with n degrees of freedom : PMM vol. 36, n≗1, 1972, pp. 125–129

We have obtained certain theorems on the linear integrals of a mechanical system with n degrees of freedom and on their dependency on the solutions of the Killing equations and the coordinates of the generalized force.

Classification of linear integrals of a holonomic mechanical system with two degrees of freedom: PMM vol. 35. n≗3, 1971, pp. 420–422

The linear integrals of a mechanical system are classified according to the solutions of the Killing equation and of the form of the generalized forces. An example of a mechanical system with two degrees of freedom which has a generalized force function but no linear integral, was given in [1].

A group-variational procedure for finding first integrals of dynamical systems

In this work a group-variational procedure is given for finding first integrals of non-conservative dynamical systems. The existence of first integrals depends on the existence of solutions of the system of partial differential equations—generalized Killing's equations. Two examples are analyzed in detail.

The generalised killing equation in finsler space

The infinitesimal deformation of a Finsler space has been studied [3, 4, 5](1) in the general form taking the vector, along which the deformation is considered, as the function of line-elements. The necessary and sufficient condition for the infinitesimal change to become a motion has been obtained in the present paper and is termed as generalised Killing equation. Various alternative forms of this equation which are generalisations of those of Hokari [1], Knebelman [2] and Soo’s [8] have been obtained.

The Killing vector and the generalised Killing equation in Finsler space

Recently we have studied the infinitesimal deformation in Finsler space [8] (1). Here we have obtained the necessary and sufficient condition for the infinitesimal change to become a motion in Finsler space. The condition is termed as the generalised Killing equation and the vector\(v^i \left( {x, \dot x} \right)\) occurring in this motion has been called the Killing vector. Various alternative forms of this equation have also been obtained. These forms are the generalisations of the corresponding equations obtained by Hokari [5], Knebelman [6] and Soos [10].