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.
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.
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.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.