We examine the classical dynamics of multifield inflation models with quadratic potentials .
Such models are shown to have inflationary attractors in phase space , consistent with the stretching of phase space trajectories along the volume factor of the universe during inflation .
Using the symplectic structure associated with Hamiltonian systems we form a measure on the phase space , as initially proposed by Gibbons , Hawking and Stewart .
This is used to calculate lower bounds on the probabilities of observational agreement ( i.e .
the probability the model gives a value for the spectral index within the region n _ { s } = 0.968 \pm { 0.006 } ) for equal mass two and three field models with quadratic potentials , giving values of 0.982 and 0.997 respectively .
We derive the measure for a general N -field model and argue that as the number of fields approaches infinity , the probability of observational agreement approaches one .