Show that the initial conditions of the generalized coordinates \(q_{i}(t)\) can be expressed in terms of those of the physical coordinates \(x_{i}(t)\) in modal analysis as

\[\vec{q}(0)=[X]^{T}[m] \vec{x}(0), \quad \dot{\vec{q}}(0)=[X]^{T}[m] \dot{\vec{x}}(0)\]