Continuum Mechanics/Coordinates

Systems of coordinates

1. The system of coordinates is a mapping from manifold to area in Euclidean space. Under map we understand bijective smooth to both sides function, i.e. diffeomorphism.
2. Jacobian is always unequal to zero.
3. Coordinate curve
4. Examples of curvilinear coordinates on manifolds: polar on 2-dimensional plane, spherical on 3-dimensional space, stereographic on 2-dimensional sphere; toroidal? elliptic? hyperbolic?; Cartesian.
5. Let $x,y,z$ always be coordinates of points in Cartesian system and $x^{1},x^{2},x^{3}$ - coordinates of a point in arbitrary curvilinear system of coordinates (there may be any other letter instead of $x$ however).

Lagrangian description

Law of motion

1. Law of motion of a point
2. Motion of continuum
3. Individualization of points of continuum
4. Law of motion of continuum
5. Lagrangian variables

Properties of functions describing law of motion

1. Continuity
2. Example: water spray (discontinuity)
3. Bijectivity; Jacobian
4. Common properties of homeomorphisms

Again about coordinate systems

1. Observer reference frame
2. Galileo principle of relativity
3. Reference frame bound to continuum
4. Essence of Lagrangian description

Stationary Spatial Coordinates

Stationary spatial coordinates are based on some stationary reference, like the building you are in. We use x,y,z when talking about stationary spatial coordinates.

Material Coordinates

With material coordinates, a certain reference state is chosen at time 0 (t=0). Each point in the material is assigned a label, which is its location at t=0. We use X,Y,Z when talking about material coordinates.

For example, suppose that g(X,Y,Z,t) tells you the temperature of an object in material coordinates. g(2,3,1,5) would be the temperature of the particle labeled (2,3,1) at t=5 which means that at t=0, the particle as at (2,3,1) in stationary spatial coordinates.

Material Derivative Example: The heat along a one dimensional fluid is defined as f(x,t). f is a function of stationary spacial coordinates, as indicated by the lower case letter x.

Let's consider what g(X,t) would be if it was also a description of the temperature of the same fluid but in material coordinates, as indicated by the upper case letter X.

g(X, t) = f(x, t)

For this to equation to have meaning, we needed to describe the relationship between x and X. Consider the x to be function of X and t, it is the location of the particle with label X at a particular time t.

x(X,t)

Note that x is dependent on time, t.

The material derivative of the temperature is a description of how the temperature changes at this one particular particle labeled X, (the particle moves as time passes). We want to take the partial derivative of g(X,t) with respect to t.

Considering x is dependent on t and using the chain rule, df/dt = df/dx dx/dt + df/dt dt/dt = df/dx dx/dt + df/dt