Abstract

Hamiltonian mechanics plays an important role in the development of nonlinear science. This paper aims for a fractional Hamiltonian system of variable order. Several issues are discussed, including differential equation of motion, Noether symmetry, and perturbation to Noether symmetry. As a result, fractional Hamiltonian mechanics of variable order are established, and conserved quantity and adiabatic invariant are presented. Two applications, fractional isotropic harmonic oscillator model of variable order and fractional Lotka biochemical oscillator model of variable order are given to illustrate the Methods and Results.

Highlights

  • Fractional calculus was firstly proposed because the order of a function’s integral or derivative is a non-integer constant

  • From Equations (9) and (10), the Riemann–Liouville fractional derivative of variable order (RVO), the Riesz–Caputo fractional derivative of variable order (RCVO), the left and right Riemann–Liouville fractional derivatives (RL), the left and right Caputo fractional derivatives (C), the Riesz–Riemann–Liouville fractional derivative (R), the Riesz–Caputo fractional derivative (RC), the combined Riemann–Liouville fractional derivative (CRL), the combined Caputo fractional derivative (CC), and the classical integer derivative can be deduced as special cases

  • Noether symmetry and conserved quantity will be discussed in detail with the CRLVO and the Caputo fractional derivative of variable order (CCVO), respectively

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Summary

Introduction

Fractional calculus was firstly proposed because the order of a function’s integral or derivative is a non-integer constant. Samko and Ross [14] introduced a generalization of fractional calculus They considered the order of a function’s integral or derivative as α(·, ·), where α(·, ·) is a function rather than a constant. The study of the fractional Noether symmetry and fractional conserved quantity with variable order has begun. Some important results on the study of perturbation to Noether symmetry and adiabatic invariants for constrained mechanical systems have been obtained [60,61]. The purpose of this paper is to generalize the problem of the calculus of variations, Noether theory, and perturbation to Noether symmetry to the fractional Hamiltonian systems in terms of combined. Noether symmetry and conserved quantities for the fractional Hamiltonian systems with the CRLVO and the CCVO are presented.

Combined Fractional Derivative of Variable Order
Hamilton Equation of Variable Order
Noether Symmetry and Conserved Quantity
Noether Symmetry and Conserved Quantity with the CRLVO
Noether Symmetry and Conserved Quantity with the CCVO
Some Explanations
For the Hamiltonian system with the RVO if there exists a gauge function
For the Lagrangian
Perturbation to Noether Symmetry and Adiabatic Invariants
Applications
Conclusions
Full Text
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