We show that on a particular class of semi-direct sums of matrix Lie algebras, component traces of the matrix product can produce bilinear forms which are non-degenerate, symmetric and invariant under the Lie product. The corresponding variational identities are called componenttrace identities and provide tools in generating Hamiltonian structures of integrable couplings including the perturbation equations. An illustrative example of applying component-trace identities is given for the KdV hierarchy.

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In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann—Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

The perturbation of symmetries and adiabatic invariants for Lagrangian system are studied. The exact invariants introduced by the Lie symmetries of Lagrangian system without perturbation are given. Based on the definition of high-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for Lagrangian system with the action of small disturbance is investigated, and a type of Hojman adiabatic invariants of the system are obtained. An example is given to illustrate the application of the results.

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