Editing Continuum mechanics (section)

===Lagrangian description===
In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time. In this case '''the reference configuration is the configuration at <math>t=0</math>'''. An observer standing in the frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. The results obtained are independent of the choice of initial time and reference configuration, <math>\kappa_0(\mathcal B)</math>. This description is normally used in [[solid mechanics]].

In the Lagrangian description, the motion of a continuum body is expressed by the mapping function <math>\chi(\cdot)</math> (Figure 2),

:<math>\mathbf x=\chi(\mathbf X, t)</math>

which is a mapping of the initial configuration <math>\kappa_0(\mathcal B)</math> onto the current configuration <math>\kappa_t(\mathcal B)</math>, giving a geometrical correspondence between them, i.e. giving the position vector <math>\mathbf{x}=x_i\mathbf e_i</math> that a particle <math>X</math>, with a position vector <math>\mathbf X</math> in the undeformed or reference configuration <math>\kappa_0(\mathcal B)</math>, will occupy in the current or deformed configuration <math>\kappa_t(\mathcal B)</math> at time <math>t</math>. The components <math>x_i</math> are called the spatial coordinates.

Physical and kinematic properties <math>P_{ij\ldots}</math>, i.e. thermodynamic properties and flow velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and time, i.e. <math>P_{ij\ldots}=P_{ij\ldots}(\mathbf X,t)</math>.

The material derivative of any property <math>P_{ij\ldots}</math> of a continuum, which may be a scalar, vector, or tensor, is the time rate of change of that property for a specific group of particles of the moving continuum body. The material derivative is also known as the ''substantial derivative'', or ''comoving derivative'', or ''convective derivative''. It can be thought as the rate at which the property changes when measured by an observer traveling with that group of particles.

In the Lagrangian description, the material derivative of <math>P_{ij\ldots}</math> is simply the partial derivative with respect to time, and the position vector <math>\mathbf X</math> is held constant as it does not change with time. Thus, we have

:<math>\frac{d}{dt}[P_{ij\ldots}(\mathbf X,t)]=\frac{\partial}{\partial t}[P_{ij\ldots}(\mathbf X,t)]</math>

The instantaneous position <math>\mathbf x</math> is a property of a particle, and its material derivative is the ''instantaneous flow velocity'' <math>\mathbf v</math> of the particle. Therefore, the flow velocity field of the continuum is given by

:<math>\mathbf v = \dot{\mathbf x} =\frac{d\mathbf x}{dt}=\frac{\partial \chi(\mathbf X,t)}{\partial t} </math>

Similarly, the acceleration field is given by

:<math>\mathbf a= \dot{\mathbf v} = \ddot{\mathbf x} =\frac{d^2\mathbf x}{dt^2}=\frac{\partial^2 \chi(\mathbf X,t)}{\partial t^2} </math>

Continuity in the Lagrangian description is expressed by the spatial and temporal continuity of the mapping from the reference configuration to the current configuration of the material points. All physical quantities characterizing the continuum are described this way. In this sense, the function <math>\chi(\cdot)</math> and <math>P_{ij\ldots}(\cdot)</math> are single-valued and continuous, with continuous derivatives with respect to space and time to whatever order is required, usually to the second or third.