On the Form-Invariance of Lagrangian Function for Higher Gradient Continuum

In this work, we consider an elastic continuum of third grade. For the sake of simplicity, we do not consider kinetic energy in the Lagrangian function. In this work, we reformulate the problem by considering Lagrangian function depending on the metric tensor

$${\mathbf g}$$

and on the affine connection

$$\nabla$$

assumed to be compatible with the metric

$${\mathbf g}$$

, and rewrite the Lagrangian function as

$${\fancyscript{L}} ({\mathbf g}, \nabla, \nabla^2).$$

Following the method of Lovelock and Rund, we apply the form-invariance requirement to the Lagrangian

$${\fancyscript{L}}.$$

It is shown that the arguments of the function

$${\fancyscript{L}}$$

are necessarily the torsion

$$\aleph$$

and/or the curvature

$$\Re$$

associated with the connection, in addition to the metric

$${\mathbf g}.$$

The following results are obtained: (1)

$${\fancyscript{L}} ( {\mathbf g}, \nabla )$$

is form-invariant if and only if

$${\fancyscript{L}} ( {\mathbf g}, \aleph );$$

(2)

$${\fancyscript{L}} ( {\mathbf g}, \nabla^2 )$$

is form-invariant if and only if

$${\fancyscript{L}} ( {\mathbf g}, \Re );$$

and (3)

$${\fancyscript{L}} ( {\mathbf g}, \nabla, \nabla^2 )$$

is form-invariant if and only if

$${\fancyscript{L}} ( {\mathbf g}, \aleph, \Re ).$$