Hamilton, Lie and the Erlangen program: the great unifications
The problems of designing and operating systems had moved from practice to a theoretical level by the mid-19th century, in the wake of the industrial revolution. This theoretical framework, which has become very topical for the modern world of systems, is summarised below.
The Hamiltonian approach is a long but consistent continuation of the Newtonian interpretations of motion. This logical process was furthered by the greatest mathematicians and physicists of three centuries. The fundamental question was the equilibrium-symmetry hypothesis relating to the causes, processes and directions of changes. From our perspective the next big step in this long creative process was the formulation of the Lagrangian equilibrium condition, bearing in mind the evolution of the problem of finding the optimal path starting with the brachystochron problem and ending with the variational calculus, and the contributions made by the Bernoulli brothers, Newton, Huygens, Euler and Gauss.
The Hamiltonian system combines three related group of questions. The first is the general conceptual category of system variables, including the H variable of the energy and potential type scalar, and the relation between multidimensional variables generalising location, momentum and time coordinates. The relations are represented by the partial derivatives corresponding to the variables. This generalisation made it possible later to use the Hamiltonian functions in quantum theory which was not even imagined at the time the functions were formulated.
(Note that the Hamiltonian H and the H of Hardy spaces used in the forms of H2 and Hinf refer to different concepts; but as the different disciplines evolved they canonised these signs for their own purposes, so we use italics to refer to the Hamiltonian H.)
The second great achievement of the unification is the expression of equilibriums. This is included in the closed summary of the Hamiltonian functions within a system, that is the generalisation of the equilibriums between potential and kinetic energy.
The third concept is the basis of calculating the direction of processes, the equilibrium dynamics working in time and other coordinates.
The Hamiltonian approach is closely related to group theory and its geometric interpretation. In the multidimensional coordinate system describing the phenomena, coordinate transformations within a given group give results which are equivalent in terms of their content. This is applied by the relativity theorem in physics and by the optimal and robust control of technological processes.