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Hunter–Saxton Equation
In mathematical physics, the Hunter–Saxton equation : (u_t + u u_x)_x = \frac \, u_x^2 is an integrable PDE that arises in the theoretical study of nematic liquid crystals. If the molecules in the liquid crystal are initially all aligned, and some of them are then wiggled slightly, this disturbance in orientation will propagate through the crystal, and the Hunter–Saxton equation describes certain aspects of such orientation waves. Physical background In the models for liquid crystals considered here, it is assumed that there is no fluid flow, so that only the ''orientation'' of the molecules is of interest. Within the elastic continuum theory, the orientation is described by a field of unit vectors n(''x'',''y'',''z'',''t''). For nematic liquid crystals, there is no difference between orienting a molecule in the n direction or in the −n direction, and the vector field n is then called a ''director field''. The potential energy density of a director field is usually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
