The Lie symmetries and conserved quantities of constrained mechanical systems are studied. Using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the systems are established. The structure equation and the form of conserved quantities are obtained. We find the corresponding conserved quantity from a known Lie symmetry, that is a direct problem of the Lie symmetries. And then, the inverse problem of the Lie symmetries-finding the corresponding Lie symmetry from a known conserved quantity-is studied. Finally, the relation between the Lie symmetry and the Noether symmetry is given.

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Symmetries and conserved quantities of constrained mechanical systems

This paper presents the variational problems for fractional Birkhoffian systems, and its corresponding symmetries and conserved quantities are further studied. First, the fractional Pfaff variational problems in terms of Riemann–Liouville fractional derivatives are proposed, and the fractional Pfaff–Birkhoff–d’Alembert principle is established. Thus, the fractional Birkhoff’s equations are derived. Second, we derive two basic formulae for the variation of Pfaff action and give the definitions and criteria of fractional Noether symmetries and quasi-symmetries. Third, we establish the Noether theorems of fractional Birkhoffian systems which reveal the relationship between fractional symmetries and fractional conserved quantities. Some special cases are discussed, showing the universality of our results. Two examples are analyzed to illustrate the applications of the results in detail.

Noether symmetries and conserved quantities for fractional Birkhoffian systems

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff–Birkhoff principle, and give four kinds of fractional Pfaff–Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff–Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman–Urrutia model, the fractional Henon–Heiles model and the fractional Emden model.

Fractional Birkhoffian mechanics

The fractional Pfaffian variational problems and the fractional Noether theory are studied under a fractional model presented by El-Nabulsi. Firstly, the fractional action-like Pfaffian variational problem is presented, the El-Nabulsi–Pfaff–Birkhoff–d’Alembert fractional principle is established, then the El-Nabulsi–Birkhoff fractional equations are derived; secondly, the definitions and criteria of the fractional Noether symmetric transformations are given, which are based on the invariance of El-Nabulsi–Pfaffian action under the infinitesimal transformations of group, then the inner relationship between a fractional Noether symmetry and a fractional conserved quantity is established; finally, two examples are given to illustrate the application of the results.

Symmetries and conserved quantities for fractional action-like Pfaffian variational problems

Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay

Birkhoffian formulations of nonholonomic constrained systems

Traditionally there do not exist integralinvariants for a nonconservative system in the phasespace of the system. For weak nonconservative systems,whose dynamical equations admit adjoint symmetries, there exist Poincare and Poincare-Cartanintegral invariants on an extended phase space, wherethe set of dynamical equations and their adjointequations are canonical. Moreover, integral invariantsalso exist for pseudoconservative dynamical systemsin the original phase space if the adjoint symmetriessatisfy certain condtions.

Poincare-Cartan Integral Invariants of Nonconservative Dynamical Systems

Usually there does not exist an integral invariant of Poincare-Cartan's type for a nonholonomic system because a constraint submanifold does not admit symplectic structure in general. An integral variant of Poincare-Cartan's type, depending on the nonholonomy of the constraints and nonconservative forces acting on the system, is derived from D'Alembert-Lagrange principle. For some nonholonomic constrained mechanical systems, there exists an alternative Lagrangian which determines the symplectic structure of a constraint submanifold. The integral invariants can then be constructed for such systems.

Poincaré-Cartan Integral Variants and Invariants of Nonholonomic Constrained Systems

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Birkhoffian formulations of nonholonomic constrained systems

Poincare-Cartan Integral Invariants of Nonconservative Dynamical Systems

Poincaré-Cartan Integral Variants and Invariants of Nonholonomic Constrained Systems