How can we tell two actions give the same equations of motion?

[Max]

The equations of motion for the two-dimensional isotropic harmonic
oscillator
ddX + a*X = 0 and ddY + a*Y = 0
can be obtained from the Lagrangian
L1(X, Y, dX, dY) = m*(dX*dY - a*X*Y)
or from
L2(X, Y, dX, dY) = m/2*(dX*dX + dY*dY - a*X*X - a*Y*Y)
where dX, ddX are first and second differential of X with time t.

The interesting part is that L1 can not be written as L2 + dF(x, y,
t)/dt. It means that the difference of the two actions can not be
exclusively assigned to boundary conditions.

In general, how can we tell two actions give the same equations of
motion?

-Max