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Mind Your  p's  and  q's!

Earlier we introduced the notion of canonical coordinates  q_i  and the generalized forces  Q_i  defined by the partial derivatives of the potential energy  V  with respect to  q_i. I promised then that I would describe a more general prescription later. Well, here it comes!

If we may assume that both the potential energy   $V(q_i,\dot{q}_i)$  and the kinetic energy   $T(q_i,\dot{q}_i)$  are known as explicit functions of the canonical coordinates  q_i  and the associated "canonical velocities"  $\dot{q}_i$,  then it is useful to define the Lagrangian function

$${\cal L}(q_i, \dot{q}_i) \; \equiv \; T(q_i,\dot{q}_i) \; - V(q_i,\dot{q}_i)$$ (12.12)

in terms of which we can then define the canonical momenta

$$p_i \; \equiv \; {\partial {\cal L} \over \partial \dot{q}_i}$$ (12.13)

These canonical momenta are then guaranteed to "act like" the conventional momentum  $m \dot{x}$  in all respects, though they may be something entirely different.

How do we obtain the equations of motion in this new "all-canonical" formulation? Well, HAMILTON'S PRINCIPLE declares that the motion of the system will follow the path  q_i(t)  for which the "path integral" of  ${\cal L}$  from initial time  t₁  to final time  t₂,

$${\cal I} \; = \; \int_{t_1}^{t_2} \; {\cal L} \; dt$$ (12.14)

is an extremum [either a maximum or a minimum]. There is a very powerful branch of mathematics called the calculus of variations that allows this principle to be used^12.5 to derive the LAGRANGIAN EQUATIONS OF MOTION,

$$\dot{p}_i \; = \; {\partial {\cal L} \over \partial q_i}$$ (12.15)

Because the "q" and "p" notation is always used in advanced Classical Mechanics courses to introduce the ideas of canonical equations of motion, almost every Physicist attaches special meaning to the phrase, "Mind your  p's  and  q's." Now you know this bit of jargon and can impress Physicist friends at cocktail parties. More importantly, you have an explicit prescription for determining the equations of motion of any system for which you are able to formulate analogues of the potential energy  V  and the kinetic energy  T.

There is one last twist to this canonical business that bears upon greater things to come. That is the procedure by which the description is re-cast in a form which depends explicitly upon  q_i  and  p_i,  rather than upon  q_i  and  $\dot{q}_i$. It turns out that if we define the Hamiltonian function

$${\cal H}(q_i, p_i) \; \equiv \; \sum_i \dot{q}_i \, p_i \; - \; {\cal L}(q_i, \dot{q}_i)$$ (12.16)

then it is usually true that

$${\cal H} \; = \; T \, + \, V$$ (12.17)

- that is, the Hamiltonian is equal to the total energy of the system! In this case the equations of motion take the form

$$\dot{q}_i = {\partial {\cal H} \over \partial p_i} \qquad \ . . . . . . \qquad \dot{p}_i = - {\partial {\cal H} \over \partial q_i}$$ (12.18)

So what? Well, we aren't going to crank out any examples, but the Lagrangian and/or Hamiltonian formulations of Classical Mechanics are very elegant (and convenient!) generalizations that let us generate equations of motion for problems in which they are by no means self-evident. This is especially useful in solving complicated problems involving the rotation of rigid bodies or other problems where the motion is partially constrained by some mechanism [usually an actual machine of some sort]. It should also be useful to you, should you ever decide to apply the paradigms of Classical Mechanics to some "totally inappropriate" phenomenon like economics or psychology. First, however, you must invent analogues of kinetic energy  V  and potential energy  T  and give formulae for how they depend upon your canonical coordinates and velocities or momenta.

Note the dramatic paradigm shift from the force and mass of Newton's SECOND LAW to a complete derivation in terms of energy in "modern" Classical Mechanics. It turns out that this shift transfers smoothly into the not-so-classical realm of QUANTUM MECHANICS, where the HAMILTONIAN  ${\cal H}$  takes on a whole new meaning.