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For a bounded domain $\Omega\subset {\Bbb R}^n$ with smooth boundary, weexplicitly calculate the first two coefficients of the asymptotic expansion ofthe trace of the strongly continuous semigroup associated with theNavier-Lam\'{e} operator on $\Omega$ as $t\to 0^+$. These coefficients (i.e.,spectral invariants) provide precise information for the volume of the elasticbody $\Omega$ and the surface area of the boundary $\partial \Omega$ in termsof the spectrum of the Navier-Lam\'{e} problem. As an application, we show thatan $n$-dimensional ball is uniquely determined by its Navier-Lam\'{e} spectrumamong all bounded elastic body with smooth boundary.

The vibrational frequencies of the elastic body and its geometric quantities