User:Tohline/SSC/Perspective Reconciliation

Reconciling Eulerian versus Lagrangian Perspectives

Linearizing the Key Relations

Continuity Equation
Lagrangian Perspective	Eulerian Perspective
<math>~\frac{d\rho}{dt}</math> <math>~=</math> <math>~- \rho \nabla\cdot\vec{v}</math>	<math>~\frac{\partial\rho}{\partial t}</math> <math>~=</math> <math>~- \rho \nabla\cdot\vec{v} - \vec{v}\cdot \nabla\rho</math>
Spherically Symmetric Initial Configurations & Purely Radial Perturbations
<math>~\frac{d\rho}{dt}</math> <math>~=</math> <math>~- \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr)</math>	<math>~\frac{\partial\rho}{\partial t}</math> <math>~=</math> <math>~- \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr) - v_r \frac{\partial \rho}{\partial r}</math>

See Also

• Eulerian Perspective
• Lagrangian Perspective