  The effect of uniaxial strain on electronic structure and magnetism in
LaMnO$_3$ is studied from a model Hamiltonian that illustrates the competition
between the Jahn-Teller, super exchange, and double exchange interactions. We
retain in our model the three main octahedral distortions ($Q_1, Q_2$, and
$Q_3$), which couple to the Mn $(e_g)$ electrons. Our results show the ground
state to be a type A antiferromagnetic (AFM) insulating state for the
unstrained case, consistent with experiments. With tensile strain (stretching
along the c axis), the ground state changes into a ferromagnetic and eventually
into a type G$^\prime$ AFM structure, while with compressive strain, we find
the type A switching into a type G structure. The orbital ordering, which
displays the well known checkerboard $x^2-1 / y^2-1$ structure for the
unstrained case, retains more or less the same character for compressive
strain, while changing into the $z^2-1$ character for tensile strains. While
$Q_1$ and $Q_3$ are fixed by the strain components $\varepsilon_{xx}$ and
$\varepsilon_{zz}$ in our model, the magnitude of the in-plane distortion mode
(Q$_2$), which varies to minimize the total energy, slowly diminishes with
tensile strain, completely disappearing as the FM state is entered. Within our
model, the FM state is metallic, while the three AFM states are insulating.
