The Euler-Lagrangian paradigm remains a powerful approach to CM. However, a crucial development occurred in 1834, when Hamilton published an alternative approach, one that had immense influence on the further development of CM, the discovery of QM, and our understanding of time.
In Hamilton’s approach, the 2n variables (q, q) used to construct the Lagran- gian are first replaced by an equivalent set of variables (q, p), where the p, are known as conjugate momenta. These are obtained from the Lagrangian by the rule
Whereas the original variables (q, q) can be thought of as points in configuration- velocity space, or some variant concept such as fibre bundles [Schutz, 1980; Abraham & Marsden, 2008], (q, p) is now regarded as the coordinates of a point in phase space, a 2n-dimensional space.
Assuming that the relationships (14.5) can be inverted or reorganized so that the velocities can be found as explicit functions of the phase-space coordinates, the next step is to define the Hamiltonian H (p, q, t), a time-dependent function over phase space, by the rule
At this point, all notion of trajectory has disappeared. It reappears with Hamilton’s equations of motion, which are given by
where the ‘dynamical equality’ = denotes an equality holding over a true (i.e.,
classical) phase-space trajectory.
Several comments are relevant here.
- 1. The inversion of the relations (14.5) is easy for most applications in Newtonian mechanics. However, that is not the case in relativistic particle mechanics and, indeed, for all modern theories of space, time, and matter including GR. This fact is behind the so-called ‘problem of time’ in quantum cosmology, which has led some theorists such as Barbour to assert that ‘time does not exist’ [Barbour, 1999]. Dirac developed a formalism called constraint mechanics to deal with this inversion issue [Dirac, 1964] because he wanted to quantize GR. We shall have more to say on this topic in the next chapter.
- 2. The Hamiltonian is related to the concept of energy, but is more subtle. Consider an SUO consisting of a bead on a smooth wire that is forced to rotate at constant angular speed in a plane about a fixed point. The Hamiltonian for this SUO is constant over any dynamical trajectory37 but the ‘energy’ is not. Such examples suggest that energy can involve the interface between endophysics and exophysics. In the case of the rotating bead, energy from outside the SUO is absorbed by the bead as it is pushed further and further out by inertial forces.
- 3. Given a phase space and a Hamiltonian, we can evaluate Hamilton’s equations of motion at any point in that phase space. These are first-order in time differential equations. Therefore, given a starting point in that phase space, these equations can transport the state of the SUO unambiguously and without further information from that point. It is almost as if the Hamiltonian gives us a set of fingerprints or grooves in phase space, the paths that would be followed by particles subject to the equations of motion. Moreover, different paths would never cross.