# Conservation Laws

• L. R. Rakotomanana
Part of the Progress in Mathematical Physics book series (PMP, volume 31)

## Abstract

The basic roots and meaning of conservation laws are reminded in Appendix B in relation with the concept of the invariance group. The scope of this book is rather limited to classical thermomechanics laws. The basic problem in deriving the conservation laws focuses on calculating the evolution of any p-form field during a continuum deformation. For the sake of simplicity, consider a continuum B of dimension three, with an affine connection ∇, a metric g, and a volume form ω, in motion with respect to a Galilean reference. The motion is defined by the application φt-t0 the velocity field of which is transcribed as v(M, t).

## Keywords

Volume Form Helmholtz Free Energy Boundary Action Exterior Derivative Integral Invariance
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## References

1. 1.
The usual version of Cauchy’s theorem starts with a scalar-valued flux. Let there be three quantities pe, re, and Je respectively of class C1,C0, and C1 verifying the conservation laws. Assume that the flux Je is a function of the unit normal vector n at ∂B: Je = Je(n). Then a unique vector field Je exists such that (this vector is called the “flux density vector”) Je(n) = Je.n.Google Scholar
2. 2.
Cauchy’s theorem (3.12) implies that there exists a vector field JC (contact force intensity) such that ωC=h0(JC) and consequently dh0(JC) = div JC ω0. Moreover, it can be shown that JC = v · σ. The exterior derivative of the mechanical action density is written: $$d\omega _C = dh_0 (p_n^i ) \otimes e_i = div (p_n^i )\omega _0 \otimes e_i .$$ Google Scholar