Flexoelectric effect, which is defined as strain gradient–induced polarization or electric gradient–induced strain in crystalline solids, can be presented as a fourth-rank tensor. The symmetry of the flexoelectric coefficients in matrix form is studied. The results indicate that the direct flexoelectric coefficients should be presented in 3 × 18 form and the converse flexoelectric coefficients in 6 × 9 form, rather than 6 × 6 form, like elastic constants. In addition, non-zero and independent elements in the matrices have been calculated for 32 point groups and 7 Ci groups. These results will provide valuable reference to the theoretical and application studies of flexoelectric effect.

No abstract

The new types of conserved quantities, which are directly induced by Lie symmetry of nonholonomic mechanical systems in phase space, are studied. Firstly, the criterion of the weak Lie symmetry and the strong Lie symmetry are given. Secondly, the conditions of existence of the new type of conserved quantities induced by the weak Lie symmetry and the strong Lie symmetry directly are obtained, and their form is presented. Finally, an Appell–Hamel example is discussed to further illustrate the applications of the results.

In this paper, a new type of conserved quantity induced directly from the Mei symmetry for a relativistic nonholonomic mechanical system in phase space is studied. The definition and the criterion of the Mei symmetry for the system are given. The conditions for the existence and form of the new conserved quantity are obtained. Finally, an example is given to illustrate the application of the result.

This paper studies the Lie symmetry and Hojman conserved quantity of the Nambu system. The determining equations of Lie symmetry for the system are given. The conditions for existence and the form of the Hojman conserved quantity led by the Lie symmetry for the system are obtained. Finally, an example is given to illustrate the application of the results.

Conformal invariance and a new type of conserved quantities of mechanical systems with variable mass in phase space are studied. Firstly, the definition and determining equation of conformal invariance are presented. The relationship between the conformal invariance and the Lie symmetry is given, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry under the infinitesimal transformations is provided. Secondly, a new type of conserved quantities of the conformal invariance are obtained by using the Lie symmetry of the system. Lastly, an example is given to illustrate the application of the results.

This paper studies the new type of conserved quantity which is directly induced by Mei symmetry of nonholonomic systems in terms of quasi-coordinates. A coordination function is introduced, and the conditions for the existence of the new conserved quantities as well as their forms are proposed. Some special cases are given to illustrate the generalized significance of the new type conserved quantity. Finally, an illustrated example is given to show the application of the nonholonomic system's results.

By introducing the coordination function f, the generalized Mei conserved quantities for the nonholonomic systems in terms of quasi-coordinates are given. Then based on the concept of adiabatic invariant, the perturbation to Mei symmetry and the generalized Mei adiabatic invariants for nonholonomic systems in terms of quasi-coordinates are studied.