Theg-Based Jordan Algebra and Lie Algebra Formulations of the Maxwell Equations

Abstract

When it is usually using a bigger algebra system to formulate the Maxwell equations, in this paper we consider a real four-dimensional algebra to express the Maxwell equations without appealing to the imaginary number and higher dimensional algebras. In terms of g-based Jordan algebra formulation the Lorentz gauge condition is found to be a necessary and sufficient condition to render the second pair of Maxwell equations, while the first pair of Maxwell equations is proved to be an intrinsic algebraic property. Then, we transform the g-based Jordan algebra to a Lie algebra of the dilation proper orthochronous Lorentz group, which gives us an incentive to consider a linear matrix operator of the Lie type, rendering more easy to derive the Maxwell equations and the wave equations. The new formulations fully match the requirements for the classical electrodynamic equations and the Lorentz gauge condition. The mathematical advantage of our formulations is that they are irreducible in the sense that, when compared to the formulations which using other bigger algebras (e.g., biquaternions and Clifford algebras), the number of explicit components and operations is minimal. From this aspect, the g-based Jordan algebra and Lie algebra are the most suitable algebraic systems to implement the Maxwell equations into a more compact form.

Non-canonical Minkowski and pseudo-Riemann frames of plasticity models with anisotropic quadratic yield criteria

“…A novel formulation for the elastoplasticity with the von Mises yield criterion (30) has been developed by Liu (1999a,b, 2000), Hong (2000, 2001), Mukherjee and Liu (2003), and Liu (2001aLiu ( , 2003aLiu ( , 2004a. Then, Liu (2004c) and Liu and Chang (2004) extended these studies to the Drucker-Prager model and convex plastic model.…”

Internal symmetry groups for the Drucker–Prager material model of plasticity and numerical integrating methods

“…However, due to non-linear nature of the model equations in plasticity, the difficulties for developing exact method to solve them are involved in this issue. For its great consumption of computational time, and the efficiency and accuracy of the calculations of mechanical problems being strongly influenced by the efficiency and accuracy of constitutive-equations solving schemes, it has drawn much attention over the past several decades and has stimulated many researches in this issue to develop accurate and economic algorithms; see, for example, Nagtegaal et al (1974); Hughes (1984); Ortiz and Popov (1985); Simo and Taylor (1985); Loret and Prevost (1986); Hong and Liou (1993); Simo and Hughes (1998); Liu (1999a, 2000); B€ uttner and Simeon (2002); Auricchio and Beirão da Veiga, 2003;Hjiaj et al (2003); Mukherjee and Liu (2003) and Liu (2001aLiu ( , 2004a among many others.…”

“…On the other hand, some recent attempts to propose the integration schemes based on the internal symmetries of simple elastoplastic constitutive models also deserve a further attention. A novel formulation for elastoplasticity has been recently developed by Liu (1999a,b, 2000); Liu and Hong (2001); Mukherjee and Liu (2003), and Liu (2001aLiu ( , 2004a. These authors have explored the internal symmetry groups of the constitutive models for perfect elastoplasticity with or without considering large deformation, for bilinear elastoplasticity, for visco-elastoplasticity, for isotropic work-hardening elastoplasticity, as well as for mixed-hardening elastoplasticity to ensure that the plastic consistency condition is exactly satisfied at each time step once the computational schemes can take these symmetries into account.…”

Lie symmetries of finite strain elastic–perfectly plastic models and exactly consistent schemes for numerical integrations

“…A novel formulation for elastoplasticity has been recently developed by Liu (1999a,b, 2000), Liu and Hong (2001), Liu (2001aLiu ( , 2003, and Mukherjee and Liu (2003). These authors have explored the internal symmetry groups of the constitutive models for perfect elastoplasticity with or without considering large rotation, for bilinear elastoplasticity, for visco-elastoplasticity, for isotropic work-hardening elastoplasticity, as well as for mixed-hardening elastoplasticity to ensure that the plastic consistency condition is exactly satisfied at each time step once the computational scheme can take these symmetries into account.…”