

Newton’s laws involve forces, and forces are vectors which
are a bit messier to handle and to think about than ordinary
functions are. In the Eighteenth and early Nineteenth Centuries
physicists got the idea of reformulating the laws of motion
in terms of energy functions particularly for systems of interacting
objects for which energy is conserved. For example, suppose we have a set of objects each with three position variables and corresponding momentum variables. The momentum variable p_{xi} corresponding to x_{i }which itself is the x coordinate of the ith object, is m_{i}v_{xi} . The kinetic energy of the object i is then . If there is a potential energy of interaction between them (such as that produced by gravitational attraction, there will be a potential energy term of the form between each pair of objects, i and j. The Hamiltonian, H, of the system will then look like The equations of motion, which correspond to F = ma
in this formulation are: (The subscript d here refers to directions x, y and z.),
These equations are called Hamilton’s equations. In actuality they have the same content as Newton’s equations
in this context. Their importance lies particularly in that
quantum mechanics can be described most easily in terms of
the Hamiltonian. If we choose a function Z of the position and momentum variables here its time dependence can be computed by the chain rule as Substituting Hamilton’s equations here we get The somewhat ugly last two terms here are called “the Poisson Bracket” of Z and H, and written as {Z, H}, so that we have
Exercises:16.1 Consider the system consisting of the sun and the earth, with a potential energy between them of . Write down Hamilton’s equations for this system. 16.2 A force in the radial direction (plus or minus) is called a central force. The force on the earth implied by the example above is an example of one, if we choose the position of the sun as origin. Compute the time derivative of r_{e}v_{e} in this system for this (or any) central force. 