How do you get field solutions to Continuum Mechanics Problems?

[Cotton_Sorbet]

I can do Calculus reasonably well in the context of multi-variables and simple undergraduate problems, but its all algebraic, I don't have a geometric understanding of how it is that the solving happens. Fortunately, with the Lagrangian method, I can appresciate at least qualitatively how one takes an arbitrary physical system and expresses it as a system of differential equations.

But I'm interested in physical systems which are three dimensional by nature, where the solutions aren't single numbers corresponding to natural frequencies or time evolutions of single particles, but rather continuous functions (eg describing the flow characteristics of air through a jet engine or how a shock wave reflects off a large object, etc.).

I'm set to take General Relativity next year (only undergraduate >.<), which I've been told is similar to how advanced fluid dynamics works.

As far as I can make out, you find a Lagrangian on the field you're working with and work backward to find the field for given boundary conditions and get some finite difference setup to give you the solution.

Can anyone elabourate on this and the general method of solving continuum problems in Differential Geometry?

Any explicit calculation (eg flow around a rotating turbine (pretty please? ) would be greatly appresciated.

(PS I know Set Theory and I've done Special Relativity, which involves Minkowsky Space and the Einstein Summation Notation, but I never really got what the Levi Chivita connection was, just the properties of the Levi Civita Symbol)

 

[pat_connell]

The general method of solving continuum problems in Differential Geometry is to start by defining a Lagrangian for the system, which essentially describes the physical laws governing the system. The Lagrangian can then be used to derive the Euler-Lagrange equations, which are a set of partial differential equations (PDEs) describing the motion of the system. These equations can then be solved using various methods, such as the finite element method or the finite difference method, which involve discretizing the domain into small elements. In some cases, analytical solutions may be available.For example, in the case of a rotating turbine, one would first define a Lagrangian describing the behavior of the turbine, including the forces acting on it and the angular velocity. This Lagrangian can then be used to derive the associated Euler-Lagrange equations, which can then be solved using various numerical methods. The solution would then describe the flow characteristics of the air around the turbine.