The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford Landmark Science)

by Roger Penrose

The versatile and original Irish mathematician William Rowan Hamilton (1805–1865) – who was also responsible for the Hamiltonian circuits discussed on p. 185) – had developed this form of the theory in a way that emphasized an analogy with wave propagation. This hint of a relation between waves and particles – and the form of the Hamilton equations themselves–was highly important for the later development of quantum mechanics. I shall return to that aspect of things in the next chapter. One novel ingredient of the Hamiltonian scheme lies in the ‘variables’ that one uses in the description of a physical system. Up until now, the positions of particles were taken as primary, the velocities being simply the rate of change of position with respect to time. Recall (p. 217) that in the specification of the initial state of a Newtonian system we needed the positions and the velocities of all the particles in order that the subsequent behaviour be determinate. With the Hamiltonian formulation we must select the momenta of the particles rather than the velocities. (We noted on p. 214 that the momentum of a particle is just its velocity multiplied by its mass.) This might seem a small change in itself, but the important thing is that the position and momentum of each particle are to be treated as though they are independent quantities, more or less on an equal footing with one another. Thus one ‘pretends’, at first, that the momenta of the various particles have nothing to do with t...

Hamiltonian mechanics- Position and momenta