In this paper, we present an interpretation that the J-integral proposed as a parameter of a crack by J. R. Rice is the shape derivative of a singular point. First, we introduce the definition of the original J-integral and the results of an investigation into its physical meaning. Then, we introduce proposals for the extension of the J-integral that were published based on the physical meanings. We also introduce that K. Ohtsuka had proposed the generalized Jintegral that extended the J-integral to three dimensions in mathematics at that time. After that, Ohtsuka realized that the Hadamard formula for the derivative of a functional with respect to domain variation can be derived from the generalized J-integral, and came into contact with the author who had proposed a shape optimization method. We show that the J-integral is the shape derivative of a singular point, which we learned from this exchange. Finally, we point out some points to note regarding the integrals used in the extended J-integral.

Nonlinear lattice models have been extensively investigated to understand the fundamental properties of vibrations and wave propagation in crystal structures. Recently, intrinsic localized modes (ILMs), also known as discrete breathers (DBs), have attracted considerable interest due to their unique properties and applications in various engineering fields. In this paper, we present the concept and fundamental properties of ILMs within the framework of theoretical lattice models. Since crystal structures can be interpreted as nonlinear lattice systems, they inherently support the existence of ILMs. We also discuss recent advancements in the study of ILMs as atomic vibrations in crystals, highlighting several significant findings. Furthermore, we introduce a numerical method for obtaining solutions of moving ILMs and analyze their dynamics through molecular dynamics simulations. Additionally, we explore energy transport mechanisms facilitated by nonlinear waves and vibrations, such as nonlinear phonons and ILMs.