Chapter 11: Linear and Angular Momentum Dynamical Variables and Their Significance

We have mentioned in previous chapters the important connection between dynamical variables in phase space and the one-parameter groups of canonical transformations that they generate. This connection is a feature of the Hamiltonian form of dynamics. For example (see the discussion in Chapter 6 on page 56) a constant of motion is a function A(q, p) on phase space that generates a group of canonical transformations under which the Hamiltonian of the system does not change. We have noted in Chapter 9 that for a constrained system, the presence of first-class constraints weakens the reciprocal relationship between dynamical variables and the corresponding groups of canonical transformations. It is worthwhile to discuss in some detail the transformationtheoretical significance of two very important dynamical variables that are defined for most systems of physical interest, these being the variables of linear and angular momentum. Such a discussion serves as a model for the later treatment of other important dynamical variables (introduced by the theories of Galilean and Poincaré relativity). In Newtonian particle mechanics, the primary significance of the linear and angular momenta is that they are constants of motion for certain kinds of forces between the particles, and their forms are derived from the form of the basic equations of motion. On the other hand, it turns out that in Hamiltonian dynamics it is more natural to define these variables by means of the transformations that they generate on the basic dynamical variables, the q's and p's, of the theory. One can also appreciate this relationship from the following point of view. In discussing Lagrangian and Hamiltonian field theory in the previous chapter, we noted that the continuous three-vector x that on the one hand enumerates the continuous infinity of coordinates ϕ(x, t) and canonically conjugate momenta π(x, t), is on the other hand to be identified (in most physically interesting examples) with points in three-dimensional Euclidean space. Thus we spoke of Lagrangian and Hamiltonian densities and of the energy density. This geometrical interpretation of the label x naturally leads to important consequences for a dynamical system of fields. Euclidean space and the relations of Euclidean geometry are unchanged under a certain set of operations on points in that space; these are the translations and rotations of three-dimensional space and all these operations taken together comprise the Euclidean group. If a physical system embedded in Euclidean space is described in the Hamiltonian form of dynamics, then the effects of translations and rotations on the system must be adequately described in the mathematical framework. We see in this chapter how the linear and angular momenta are intimately connected with the translational and rotational properties for systems permitting a Lagrangian and Hamiltonian description. We find that both for multiparticle and field systems we can in fact define these dynamical variables in terms of the effects of translations and rotations. But the meaning of the algebraic relationship between the different components of linear and angular momenta, as well as the complete group-theoretical relationship between them and the Euclidean group, becomes clear in later chapters…