In the previous chapter it was mentioned that there is no general technique for solving the n coupled second-order Lagrange equations of motion, but that Jacobi had derived a general method for solving the 2n coupled canonical equations of motion, allowing one to determine all the position and momentum variables in terms of their initial values and the time.

There are two slightly different ways to solve Hamilton's canonical equations. One is more general, whereas the other is a bit simpler, but is only valid for systems in which energy is conserved. We will go through the procedure for the more general method, then solve the harmonic oscillator problem by using the second method.

Both methods involve solving a partial differential equation for the quantity S that is called “Hamilton's principal function.” The problem of solving the entire system of equations of motion is reduced to solving a single partial differential equation for the function S. This partial differential equation is called the “Hamilton–Jacobi equation.” Reducing the dynamical problem to solving just one equation is quite satisfying from a theoretical point of view, but it is not of much help from a practical point of view because the partial differential equation for S is often very difficult to solve. Problems that can be solved by obtaining the solution for S can usually be solved more easily by other means.

In this chapter we begin by considering canonical transformations. These are transformations that preserve the form of Hamilton's equations. This is followed by a study of Poisson brackets, an important tool for studying canonical transformations. Finally we consider infinitesimal canonical transformations and, as an example, we look at angular momentum in terms of Poisson brackets.

Integrating the equations of motion

In our study of analytical mechanics we have seen that the variational principle leads to two different sets of equations of motion. The first set consists of the Lagrange equations and the second set consists of Hamilton's canonical equations. Lagrange's equations are a set of n coupled second-order differential equations and Hamilton's equations are a set of 2n coupled first-order differential equations.

The ultimate goal of any dynamical theory is to obtain a general solution for the equations of motion. In Lagrangian dynamics this requires integrating the equations of motion twice. This is often quite difficult because the Lagrangian (and hence the equations of motion) depends not only on the coordinates but also on their derivatives (the velocities). There is no known general method for integrating these equations. You might wonder if it is possible to transform to a new set of coordinates in which the equations of motion are simpler and easier to integrate. Indeed, this is possible in some situations.

When the B-58 Hustler bomber entered service in 1958 it was a very futuristic looking delta wing bomber creating a lot of sensation. Intended as a successor of the B-47 Stratojet it was capable of reaching twice the speed of sound.

However, development went not without problems and costs risings went so out of control that the whole project was almost cancelled a few times. Strategic Air Command was initially against ordering the B-58 for service, not only because of its complexity but also since they saw no advantage of a Mach 2 bomber over other types. In spite of this the B-58 entered into service at S.A.C. in 1960. It would have a relatively short operational career…

Armed with nuclear weapons as a ‘deterrent force’ it was a typical product of the Cold War and the Soviet Union, at that time ‘The Enemy’ had nothing comparable. At this point the Cold War scene changed drastically in a short time. The first nuclear-armed ballistic missiles became reality and long-range anti-aircraft missiles were stationed at both sides. With extensive Early Warning systems also ‘on the other side’, the B-58 seemed to have very slight chances to penetrate enemy airspace undetected and survive. Since it was unsuitable to fly at very low altitude below the radar it had a relatively short operational career of less than ten years. Most of the B-58s still operational by the end of the sixties went to the big aircraft depositary at Davis Monthan in the Arizona desert and they were soon scrapped. They had outlived their operational usefulness by the changing situation in the world and the B-58 was not followed by a successor that could beat its performances..

EARLY DEVELOPMENT:

The design study for a supersonic medium bomber to replace the B-47 Stratojet started at Convair by late 1951 although preliminary work was already done as early as 1947. Both Convair and Boeing prepared design proposals known as Project MX-1965 from Convair and Project MX-1964 from Boeing. Convair's project showed a delta wing layout with four engines originally fitted in double nacelles beneath the wings and a sleek area-ruled fuselage. In fact, the first mock-up showed this engine layout that was later replaced by four separate engine pods.