Given fixed starting position and time, and , and fixed ending position and time, and , when a particle travels along a path of ‘least’ action, it obeys the Euler-Lagrange equations for some Lagrangian, :
which is a fancy way of saying the time derivative of (momentum) is equal to (force). The natural variables for this Lagrangian approach are (position) and (velocity), from which and are built.
Given the right conditions, we can recast this formulation into the Hamiltonian approach, where and are the natural variables, and several nice things happen. By ‘right conditions’ I mean conditions that would allow us to return to the Lagrangian approach. One of the nice things that happen is position and momentum can be seen to be conjugate to each other in a way that is impossible for position and velocity. Indeed, Hamilton’s equations illustrate this conjugacy nicely:
where is the total energy function.
If and are so similar, could we not recast this yet again, but in terms of and , much like the Lagrangian approach was in terms of and ? Or if we are feeling zealous with reformulations, why not consider mechanics recast in terms of and , like a derived Hamiltonian approach? These are some questions that occurred to me near the end of last quarter in Dr. Baez’ Classical Mechanics (Math 241) course, when he went over how to transition from the Lagrangian approach to the Hamiltonian approach.
At first blush, these ‘new’ approaches seem to give us less information than the old ones. To wit, in Newtonian mechanics, , so any information about the absolute position appears to be lost to a constant of integration, especially for the approach. But it is worse than this. For each particle being considered, the constant of integration may be different. So not only do we lose absolute position, we also lose relative position. This limits our considerations to situations where potential energy is zero, unless something very nice happens that would allow us to recover .
The nice thing about mathematics versus physics, is that I can cast aside the difficulties of whether or not things are physically relevant, as long as they make sense mathematically. So I will set aside any complaints about possible non-utility and forge ahead. I have not actually looked that far ahead yet, but I suspect the first ‘new’ approach will be somewhat similar to the Lagrangian approach, via the analogy between and . That is, I suspect there will be some function, , analogous to the Lagrangian, , such that:
where and will be interesting quantities, perhaps even position and velocity.
The second ‘new’ approach will almost certainly be lossy, but I suspect it will follow a pattern similar to the Hamilton equations. For convenience, I will write for these:
where is something analogous to total energy, and should have units of power / time.
So I leave, for now, with some things to ponder, and some guesses as to the direction they will lead.