  The present paper deals with an \emph{intrinsic} investigation of the notion
of a concurrent $\pi$-vector field on the pullback bundle of a Finsler manifold
$(M,L)$. The effect of the existence of a concurrent $\pi$-vector field on some
important special Finsler spaces is studied. An intrinsic investigation of a
particular $\beta$-change, namely the energy $\beta$-change
($\widetilde{L}^{2}(x,y)=L^{2}(x,y)+ B^{2}(x,y) with \
B:=g(\bar{\zeta},\bar{\eta})$; $\bar{\zeta} $ being a concurrent $\pi$-vector
field), is established. The relation between the two Barthel connections
$\Gamma$ and $\widetilde{\Gamma}$, corresponding to this change, is found. This
relation, together with the fact that the Cartan and the Barthel connections
have the same horizontal and vertical projectors, enable us to study the energy
$\beta$-change of the fundamental linear connection in Finsler geometry: the
Cartan connection, the Berwald connection, the Chern connection and the
Hashiguchi connection. Moreover, the change of their curvature tensors is
concluded.
  It should be pointed out that the present work is formulated in a prospective
modern coordinate-free form.
