  We have studied the volume collapse of LaMnO$_3$ at the Jahn- Teller (JT)
transition temperature T$_{JT}$=750 K which has recently been found in high
temperature powder x- ray and neutron diffraction experiments. We construct a
model Hamiltonian involving the pseudospin of Mn$^{3+}$ e$_g$ states, the
staggered JT distortion and the volume strain coordinate. We show that the
anharmonic coupling between these primary and secondary order parameters leads
to the first order JT phase transition associated with a comparatively large
reduction of the unit cell volume of $\Delta$V/V$\simeq$ 10$^{-2}$. We explain
the temperature dependence of JT distortions and volume strain and discuss the
volume change as function of the anharmonic coupling constant. A continuous
change to a second order transition as function of model parameters is
obtained. This behaviour is also observed under Ba doping.
