Fractional cyclic integrals and Routh equations of fractional Lagrange system with combined Caputo derivatives

In this paper, we develop a fractional cyclic integral and a Routh equation for fractional Lagrange system defined in terms of fractional Caputo derivatives. The fractional Hamilton principle and the fractional Lagrange equations of the system are obtained under a combined Caputo derivative. Furthermore, the fractional cyclic integrals based on the Lagrange equations are studied and the associated Routh equations of the system are presented. Finally, two examples are given to show the applications of the results.

First Integrals and Reduction of the Birkhoffian System

Refer to the Hamiltonian system, first integrals of the Birkhoffian system can be found by using of the perfect differential method. Through these first integrals, the order of the Birkhoffian system can be reduced. Then according to the alternate of the coordinate, a kind of new partial differential operator was defined in order to hold the Birkhoff form. The result shows that the Birkhoffian system has generalized energy integrals and cyclic integrals. Furthermore, each integral can reduce the order of equations two degrees.

Influence of Contractility on Myocardial Ultrasonic Integrated Backscatter and Cyclic Variation in Integrated Backscatter

To evaluate the effects of left ventricular contractility on the changes of aver age image intensity (AII) of the myocardial integrated backscatter (IB) and cyclic variation in IB (CVIB), 7 adult mongrel dogs were studied. The magnitude of AII and CVIB were measured from myocardial IB carves before and after dobuta mine or propranolol infusion. Dobutamine or propranolol did not affect the magnitude of AII (13.8±0.7 vs 14.7±0.5, P >0.05 or 14.3±0.5 vs 14.2±0.4, P >0.05). However, dobutamine produced a significant increase in the magnitude of CVIB (6.8±0.3 vs 9.5±0.6, P <0.001) and propranolol induced significant decrease in the magnitude of CVIB (7.1±0.2 vs 5.2±0.3, P <0.001). The changes of the magnitude of AII and CVIB in the myocardium have been demonstrated to reflect different myocardial physiological and pathological changes respectively. The alteration of contractility did not affect the magnitude of AII but induced significant change in CVIB. The increase of left ventricular contractility res ulted in a significant rise of the magnitude of CVIB and the decrease of left ventricular contractility resulted in a significant fall of the magnitude of CVIB.

Methods of reduction for Lagrange systems on time scales with nabla derivatives

The Routh and Whittaker methods of reduction for Lagrange system on time scales with nabla derivatives are studied.The equations of motion for Lagrange system on time scales are established, and their cyclic integrals and generalized energy integrals are given. The Routh functions and Whittaker functions of Lagrange system are constructed, and the order of differential equations of motion for the system are reduced by using the cyclic integrals or the generalized energy integrals with nabla derivatives. The results show that the reduced Routh equations and Whittaker equations hold the form of Lagrnage equations with nabla derivatives. Finally, two examples are given to illustrate the application of the results.

Routh Order Reduction Method of Relativistic Birkhoffian Systems

Routh order reduction method of the relativistic Birkhoffian equations is studied. For a relativistic Birkhoffian system, the cyclic integrals can be found by using the perfect differential method. Through these cyclic integrals, the order of the system can be reduced. If the relativistic Birkhoffian system has a cyclic integral, then the Birkhoffian equations can be reduced at least by two degrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics, and the relativistic Lagrangian mechanics are discussed, and the Routh order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of the result.

Routh method of reduction for Birkhoffian systems in the event space

For a Birkhoffian system in the event space, this paper presents the Routh method of reduction. The parametric equations of the Birkhoffian system in the event space are established, and the definition of cyclic coordinates for the system is given and the corresponding cyclic integral is obtained. Through the cyclic integral, the order of the system can be reduced. The Routh functions for the Birkhoffian system in the event space are constructed, and the Routh method of reduction is successfully generalized to the Birkhoffian system in the event space. The results show that if the system has a cyclic integral, then the parametric equations of the system can be reduced at least by two degrees and the form of the equations holds. An example is given to illustrate the application of the results.