Xanthippi Markenscoff and Shailendra Pal Veer Singh
Vol. 10 (2015), No. 3, 331–353
DOI: 10.2140/jomms.2015.10.331
Abstract
By the application of Noether’s theorem, conservation laws in linear
elastodynamics are derived by invariance of the Lagrangian functional
under a class of infinitesimal transformations. The recent work of Gupta
and Markenscoff (2012) providing a physical meaning to the dynamic
-integral as
the variation of the Hamiltonian of the system due to an infinitesimal translation of the
inhomogeneity if linear momentum is conserved in the domain, is extended here to the
dynamic
-
and
-integrals
in terms of the “if” conditions. The variation of the Lagrangian is shown to be equal
to the negative of the variation of the Hamiltonian under the above transformations
for inhomogeneities, which provides a physical meaning to the dynamic
-,
- and
-integrals
as dissipative mechanisms in elastodynamics. We prove that if linear momentum is
conserved in the domain, then the total energy loss of the system per unit scaling under
the infinitesimal scaling transformation of the inhomogeneity is equal to the dynamic
-integral,
and if linear and angular momenta are conserved then the total energy loss of the system
per unit rotation under the infinitesimal rotational transformation is equal to the dynamic
-integral.
