A linear frictional force in the equation of motion of a particle is obtained if one modifies the Lagrange equation of motion by the addition of the frictional term, which may be expressed in terms of the Rayleigh dissipative function. It is shown that one can construct Lagrangians in several simple cases so that the Lagrange equation of motion contains such a frictional force without the addition of any terms. This method is also useful in analyzing some aspects of the motion.
The partition function proposed by Feynman for liquid He 4 , based on his path integral method, is evaluated for a simple cubic lattice considering long-range permutations as well as nearest-neighbor permutations (to which the previous analysis by one of the authors was restricted). The result indicates a second-order phase transition at the X point. The marked improvements over the previous treatment are:(1) the specific heat behaves as T s near absolute zero, (2) the specific heat peak is more pronounced at the X point, and (3) when triangles are added as possible finite polygons above T\ the specific heat just above T\ increases over the previous result, showing an improvement. Equating the theoretical X point with the experimental, a value for the effective mass of a helium atom about 1.6 times the normal mass is obtained.
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