  By introducing the divergence of a vector potential into the Lagrangian, a
Lagrangian framework is developed to incorporate surface energy into
elasticity. Besides the Euler-Lagrange equation and natural boundary condition,
a new boundary constitutive equation is derived from the variation of the
Lagrangian and configuration on which the Lagrangian is defined. On the
boundary surface, explicit expression of the vector potential with respect to
field variable and surface curvature is determined. Based on this framework, an
elastic model with relevance to the surface energy is established. The
Young-Laplace's formula is generalized into elastic solid in a new form. Making
use of this model, we investigate the surface energy effect in the radial
vibration of spherical nanoparticle. Numerical calculation shows that natural
frequencies of nanoparticle will shift down due to the surface energy. This
shift is especially apparent in the vibration of soft matter nanoparticles.
