Based on the new concept of thermal mass which refers to the equivalent mass of the thermal energy in an object according to Einstein's mass-energy relation, the kinetic energy and potential energy of thermal mass as well as its dissipation were introduced to establish the Lagrange equations in heat conduction. The results show that Fourier's law is equivalent to the Lagrange equations with negligible inertia forces. The application of Lagrange equations in heat conduction makes it possible to unify the treatment of heat transfer and mechanics as well as electrics through the method of analytical mechanics. © 2009 Tsinghua University Press.
Wu, J., Guo, Z., & Song, B. (2009). Application of Lagrange Equations in Heat Conduction. Tsinghua Science and Technology, 14(SUPPL. 2), 12–16. https://doi.org/10.1016/S1007-0214(10)70023-7