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Fractional Noether's theorem in the Riesz-Caputo sense
Gastao S. F. Frederico, Delfim F. M. Torres
TL;DR
Fractional variational problems with Riesz-Caputo derivatives require a suitable notion of conservation law and a corresponding symmetry theorem. The paper develops fractional Noether theorems in Lagrangian and Hamiltonian forms, establishing links between variational symmetries and fractional conservation laws while retaining a one-variable, equal-order setting.
Problem
The paper addresses how Noether’s theorem and conservation laws should be formulated for fractional variational and optimal-control problems with Riesz-Caputo derivatives.
Method
The paper introduces an extended fractional conservation-law operator and proves Riesz-Caputo Noether theorems for variational problems and fractional optimal control in Lagrangian and Hamiltonian forms.
Results
The theorems establish fractional conservation laws along Riesz-Caputo Euler-Lagrange extremals and relate variational symmetries to those conservation laws.
Takeaways & Limitations
The results provide a fractional Noether framework for analyzing Riesz-Caputo variational and optimal-control problems and motivate adapting Leitmann’s direct method to fractional settings.
Takeaways & Limitations
The analysis is restricted to one independent variable and equal left and right fractional orders; multiple-integral and two-order extensions remain open directions.
Abstract
from arXiv · showhide
We prove a Noether's theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.
1 Introduction
The paper places fractional variational calculus within Noether’s symmetry framework and develops a Riesz-Caputo fractional Noether theorem for variational and optimal-control problems. It also identifies scope boundaries and contrasts its real-valued, one-variable, Caputo-based setting with earlier approaches.
- Variational symmetries preserve variational or optimal-control problems and are connected with conservation laws that can reduce Euler-Lagrange equation order.
- Leitmann’s direct method obtains absolute extremizers without Euler-Lagrange equations and is closely connected with Noether’s principle.
- The paper develops a fractional version of Noether’s symmetry theorem for variational problems with Riesz-Caputo derivatives, covering both calculus of variations and optimal control.
- The paper treats real-valued functions with left and right Caputo derivatives, whereas earlier work considered complex-valued functions or Riemann-Liouville derivatives.
- The analysis is restricted to one independent variable, while extending the results to multiple fractional variational integrals remains an open question.
2 Preliminaries on Fractional Calculus
This section establishes the fractional-calculus notation used throughout the paper. It collects Riemann-Liouville integrals and derivatives together with Caputo, Riesz, and Riesz-Caputo derivative definitions and their relations.
- The preliminaries define left and right Riemann-Liouville fractional integrals and the Riesz fractional integral for continuous functions on [a, b].
- They define left and right Riemann-Liouville fractional derivatives, with n selected so that n − 1 ≤ α < n and D denoting the usual derivative.
- They define left and right Caputo fractional derivatives and introduce the corresponding Riesz and Riesz-Caputo derivatives.
- The section records relations among these fractional operators, including a particular simplification for 0 < α < 1.
3 Main Results
The paper establishes Noether theorems for fractional variational and optimal-control problems using Riesz-Caputo derivatives, including fixed-time, time-reparameterized, and Hamiltonian formulations.
- 3.1 On the Riesz-Caputo conservation of momentum: The fractional conserved quantity is built with a new operator involving both Riesz and Riesz-Caputo derivatives, generalizing the classical product derivative.
- 3.1 On the Riesz-Caputo conservation of momentum: Theorem 21 establishes a fixed-time fractional Noether theorem, defining a Riesz-Caputo conservation law along fractional extremals.
- 3 Main Results: When α = 1, the fractional conservation laws reduce to the classical Noether laws, including momentum conservation and the standard optimal-control result.
- 3.2 The Noether theorem in the sense of Riesz-Caputo: Theorem 24 gives a general fractional Noether theorem for time-reparameterized Riesz-Caputo variational problems, yielding a conserved law along fractional extremals.
- 3.3 Optimal control of Riesz-Caputo fractional systems: Theorem 35 extends the result to fractional optimal control in Hamiltonian form under variational invariance.
- 3.3 Optimal control of Riesz-Caputo fractional systems: For autonomous fractional optimal control, the Hamiltonian is not conserved alone; conservation instead involves the Hamiltonian plus an α-dependent quantity.
4 Examples
The examples apply the fractional conservation laws to autonomous variational and optimal-control problems, recovering classical energy conservation when α = 1.
- 4 Examples: The paper applies Corollary 37 to two autonomous examples: one fractional calculus-of-variations problem and one fractional optimal-control problem.Both examples use Lagrangians without explicit dependence on t.
- 4 Examples: The variational example yields a fractional conservation law along the problem’s extremals.
- 4 Examples: The optimal-control example yields a conservation law along any fractional Pontryagin extremal.
- 4 Examples: For α = 1, the conservation laws from both examples reduce to the classical conservation of energy.
5 Conclusions and Possible Extensions
The conclusions present the Riesz-Caputo Noether theorem as a way to obtain conservation laws for difficult fractional variational problems, while identifying extensions to broader derivative settings.
- 5 Conclusions and Possible Extensions: Conservation laws are proposed as a way to simplify nonlinear fractional Euler-Lagrange equations that are generally difficult to solve.
- 5 Conclusions and Possible Extensions: The paper’s main contribution is a fractional Noether theorem relating variational symmetries to fractional conservation laws.The theorem applies along Euler-Lagrange extremals.
- 5 Conclusions and Possible Extensions: The results suggest that Leitmann’s direct method could be adapted to fractional variational problems, although this remains an open question.The conclusion notes that preliminary results already exist.
- 5 Conclusions and Possible Extensions: The authors emphasize that fractional dynamics depend on the derivative chosen, with no single fractional-calculus approach being universally best.
- 5 Conclusions and Possible Extensions: The theorem is restricted to Riesz-Caputo derivatives whose left and right derivatives have the same order α.Possible extensions include different left and right orders and generalized Erdélyi-Kober derivatives.