The Generalized Hamilton Principle and Non-Hermitian Quantum Theory

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the Hermitian quantum theory, i.e., the standard Schrodinger equation. In this paper, we have generalized the Hamilton principle to the generalized Hamilton principle, which can describe the open system (mass or energy exchange systems) and nonconservative force systems or dissipative systems, and given the generalized Lagrange function, it has to do with the kinetic energy, potential energy and the work of nonconservative forces to do. With the Feynman path integration, we have given the non-Hermitian quantum theory of the nonconservative force systems. Otherwise, with the generalized Hamilton principle, we have given the generalized Hamiltonian for the particle exchanging heat with the outside world, which is the sum of kinetic energy, potential energy and thermal energy, and further given the equation of quantum thermodynamics. PACS: 03.65.-w, 05.70.Ce, 05.30.Rt

Kinetic to Potential Energy Transformation Using a Spring as an Intermediary: Application to the Pole Vault Problem

The kinetic energy of a mass moving horizontally can be completely converted into potential energy using a spring as an intermediary. The spring can be used to temporarily store some of the energy of the mass and change the direction of motion of the mass from horizontal to vertical. A nondimensional framework is used to study this problem for a point mass, first with a linear spring and then with a nonlinear spring that is an elastica. Solutions to the problems with the linear spring and elastica show many similarities and some dissimilarities. The dynamics of the point mass and elastica resemble the mechanics of a pole-vault; and therefore, a nonconservative external torque is introduced to parallel the muscle work done by vaulters. For the nonconservative system, the problem is solved for complete transformation of the kinetic energy of the mass and the work done by the external torque into potential energy of the mass. The initial velocities for the two cases, with and without the nonconservative force, are quite similar; and therefore, the maximum potential energy of the mass is higher in the presence of the nonconservative force. A realistic dimensional example is considered; the solution to the problem, despite several simplifying assumptions, is found to be similar to data of elite pole vaulters presented in the literature.

CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN FOUR-DIMENSIONAL SPACE

In this article, we systemize some results on the study of the equations of motion of dynamically symmetric fixed four-dimensional rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free four-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. We also show the nontrivial topological and mechanical analogies.