[AntonioLao]

Appearing near the end of many advanced textbooks of analytical mechanics is the topic on a nonlinear Hamilton-Jacobi (HJ) partial differential equation (PDE). It highlighted theories on canonical and contact transformations. The word ‘canonical’ literally means the simplest, most symmetric, and of highest standard. Although for less fortunate investigators it becomes the most complex, most unreal, and the farthest from a norm. Furthermore, in many colleges and universities HJ PDE rarely lectured at the undergraduate or even graduate level. The excuses are always lack of time. However, two good reasons why many authors always delayed discussions till the last chapters are (1) lack of comprehensive understanding and (2) desire for continued independently self-motivated research. In this context it simply means preserving the forms of Hamilton’s canonical equations. The word ‘contact’ in accordance with Hamilton’s principle is the path independence between two fixed points at two fixed times such that the variation of the time integral of the Lagrangian is zero or stationary. The subtle differences are that the former implies invariance while the latter implies equality. The invariance is the multi-dimensional volume integral of Liouville’s theorem. The equality is the multi-dimensional path integral of Feynman’s theorem.

Feynman’s theorem suggests recasting Lagrange’s equations as initial force minus final force equal zero, Fi-Ff=0 or Fi=Ff. Liouville’s theorem suggests recasting HJ PDE as initial energy plus final energy equal zero, Ei+Ef=0 or Ei=-Ef. The 1st allows all possible values while the 2nd allows only the value of zero. Allowing all values of energy is simply to square both sides such that Ei=Ef is always true. For a quantized field theory, the variation of energy squares must satisfy the double time integrals:
H=òòdE²dtdt where dE²=dF1´dR1×dF2´dR2. When expanded by Lagrange’s identity these variations of generalized forces and generalized radius vectors give two variational topological structures: dH+=(dF1×dR2)(dF2×dR1)-(dF1×dF2)(dR2×dR1) and dH-=(dF1×dF2)(dR2×dR1)-(dF1×dR2)(dF2×dR1) whose variations are dH+=0 and dH-=0 such that dH=(dH+)+(dH-)=0. For single time integral these become true differential forms. Note that H+ and H- become real valued constants if and only if either dF1×dF2=0 or dR1×dR2=0. The latter is equivalent to the square of the space-time interval of special relativity and the former is equivalent to the Gaussian and mean curvatures of these topological structures. By matrix descriptions, quantum mechanics use dR1×dR2 to represent the square of the wave function for probability amplitude while in general relativity, dF1×dF2 is used to represent the metric tensor.