Balance, Partition Function and Hamiltonians

Often, Lagrangians yielding the same equations of motion yield the same partition function. For the case of L= U(x)U(x) + mm/12 vvvv + mU(x) vv ((1)) this does not occur. As shown in (1), ((1)) yields the same same equations of motion as H = Hamiltonian = vv/2m + U(x) with p=mv, but the Hamiltonian of ((1)) is H= [vv/2m + U(x)][vv/2m + U(x)]. It is shown in (1), that the partition functions differ. In previous notes, we have argued that the partition function may be computed through two body scattering. In particular, the Maxwell-Boltzmann distribution follows from elastic scattering .5mv1v1 + .5mv2v2 = .5mv3v3 + .5mv4v4 and f(v1)f(v2) = f(v3)f(v4), where f(v) is the distribution function. We argue that one does not need a Hamiltonian to calculate a partition function, only a conserved quantity which does not change sign if v→ -v. In such a case, the two Hamiltonians yielding the same equations of motion need not be used in the partition function calculation. What matters, we argue, are the equations of motion which may be used to establish a conserved quantity. We argue that two body scattering defines the statistical system with any potentials accelerating the particle from one point to another. In some cases, a velocity dependent potential does not cause any acceleration. At a point of collision, one simply has velocities. At early points in x, one has kinetic energy plus any potential energy terms which cause acceleration and establish conservation of energy. Thus, the Hamiltonian of ((1)) yields the same equations as H=vv/2m + U(x) where p=mv, and U(x) is the potential which together with vv/2m ensures energy conservation. From this, one would calculate the usual partition function using vv/2m + U(x) as the conserved quantity