Ip. Pavlotsky et M. Strianese, IRREVERSIBILITY IN CLASSICAL MECHANICS AS A CONSEQUENCE OF POINCARE-GROUP, International journal of modern physics b, 10(21), 1996, pp. 2675-2685
In the post-Galilean approximation of Poincare Group (i.e. in the appr
oximation in which the corrections of order O(c(-2)), c denoting the l
ight velocity, to the Galilei group are taken in account) the Lagrangi
ans are singular on a submanifold of the phase space. It is a local si
ngularity, which differs from the ones considered by Dirac. The dynami
cal properties are essentially peculiar on the studied singular surfac
es.(1-4) In particular, on some submanifolds of the singular manifold
the velocities are not determined uniquely: in each point of the subma
nifold we get the infinite set of components of velocity. It means the
loss of the reversibility of the motion in a sense that transformatio
n of the phase space, corresponding to Lagrangian, has not a property
of a group. It is shown, that if the values of the derivates of the mo
lecular potentials are large enough the irreversibility of motion take
s place. As consequence we obtain the relaxation to the equilibrium. T
his property does not exist if the Lagrangian is invariant with respec
t to Galilei Group.