Classical theory of a rigid magnetic continuum

Abstract

A consistent classical Lagrangian and Hamiltonian nonrelativistic theory of a rigid nonconducting magnetic continuum in a magnetic field is presented. The theory starts with a Lagrangian density that describes the system. The Lagrangian equation of motion for the magnetization is shown to be that which generalizes Larmor's equation to continua, namely, $\frac{d\overrightarrow{\mathrm{m}}}{\mathrm{dt}}=\gamma \overrightarrow{\mathrm{m}}\times {\overrightarrow{\mathrm{B}}}^{\mathrm{eff}}$. The transition to the Hamiltonian which describes the system is effected by means of Dirac's theory for systems with constraints. The Hamiltonian equation of motion for the time rate of change of the magnetization is identical with that obtained from the Lagrangian, as it must be. Moreover, we calculate the generalized Poission brackets (known as Dirac brackets) of the magnetization components and find ${\left\{m_i\right(\overrightarrow{\mathrm{z}},t),m_j({\overrightarrow{\mathrm{z}}}^{\prime },t\left)\right\}}^{*}=\gamma {\epsilon }_{\mathrm{ijk}}{m}_k{\delta }^3(\overrightarrow{\mathrm{z}}-{\overrightarrow{\mathrm{z}}}^{\prime })$.