Lagrangian conservation relations

Models seldom express the conservation relations in a Lagrangian framework. The chain rule of calculus is used to convert to an Eulerian framework. 

It is worth investigating the time derivatives and demonstrating how to derive (9.1)-(9.4) from the more familiar forms of the conservation equations. The more familiar Lagrangian derivative djdt and d jdt are related by [9]... 

An interesting class of exact self-similar solutions (H2) can be deduced for the case where the newly formed phase density is a function of temperature only. The method involves a transformation to Lagrangian coordinates, based upon the principle of conservation of mass within the new phase. A similarity variable akin to that employed by Zener (Z2) is then introduced which immobilizes the moving boundary in the transformed space. A particular case which has been studied in detail is that of a column of liquid, initially at the saturation temperature T , in contact with a flat, horizontal plate whose temperature is suddenly increased to a large value, Tw T . Suppose that the density of nucleation sites is so great that individual bubbles coalesce immediately upon formation into a continuous vapor film of uniform thickness, which increases with time. Eventually the liquid-vapor interface becomes severely distorted, in part due to Taylor instability but the vapor film growth, before such effects become important, can be treated as a one-dimensional problem. This problem is closely related to reactor safety problems associated with fast power transients. The assumptions made are ... 

By Noether s theorem, invariance of the Lagrangian under an infinitesimal time displacement implies conservation of energy. This is consistent with the direct proof of energy conservation given above, when L and by implication H have no explicit time dependence. Define a continuous time displacement by the transformation t = t + oi(t ) whereat/(,) = a(t ) = 0. subject to a —0. Time intervals on the original and displaced trajectories are related by dt = (1 + a )dt or dt = (1 — a )dt. The transformed Lagrangian is... 

We have now derived the four basic (time-independent) equations of stellar structure. These are mass continuity (Eq. (14)), hydrostatic equilibrium (Eq. (17)), conservation of energy (Eq. (28)), and energy transport (Eq. (33)). These form a set of coupled first order ordinary differential equations relating one independent variable, e.g. r, to four dependent variables i.e., m, /, / //, which uniquely describe the structure of the star, note that any variable could be used as the independent variable. In an Eulerian frame, the spatial coordinate r is the independent variable. For most problems in stellar structure and evolution it is usually more convenient to work in a Lagrangian frame, with mass as the independent variable. Transforming, we obtain ... 

---