# What if the Principle of Least Action were different?

1. Jan 25, 2016

### crastinus

What would the world be like if the Principle of Least Action were different? Let's say that it minimized a different quantity than KE - PE.

EF Taylor et al argue that if the quantity minimized were KE + PE, physical systems would accelerate apart from one another. Here's their short articlet; they reach this conclusion about a third of the way through (147-149): http://www.eftaylor.com/pub/ForceEnergyPredictMotion.pdf

What if PLA minimized KE/PE or KE * PE? What would the world look like in each case?

(Of course, some people think the PLA is just a neat formalism. I'm not one of them.)

2. Jan 25, 2016

### Staff: Mentor

Did you derive the corresponding equations of motion? Then check what they would give as dynamics.

3. Jan 25, 2016

Staff Emeritus
I'm not sure you even get equations of motion, as neither KE/PE nor KE*PE have units of energy.

4. Jan 25, 2016

### Staff: Mentor

You would need a principle of least energy-squared or scalar "action", but blindly appliying the rules for conventional physics should lead to some equations.

5. Jan 25, 2016

Staff Emeritus
You'll get something, but I don't think they'll necessarily be equations of motion - i.e. will not lead to an x(t).

6. Jan 25, 2016

### Hornbein

I dunno, but Greg Egan put a lot of work into figuring out what a world with an x^2+y^2+z^2+t^2 metric would be like. It's all on his web site under "orthogonal universe."

"What if PLA minimized KE/PE or KE * PE?" You would have a hard row to hoe trying to get those units to make sense. My guess is that the action has to be a bilinear operator, otherwise things start to depend on choice of units and nothing makes sense.

7. Jan 26, 2016

### crastinus

I'm not sure that we can say some set of units doesn't make sense just because they do not map to units that we understand well. In fact, one way to see the whole question is to take it as being about what the world would be like if the relevant units were different.

" . . . blindly appliying the rules for conventional physics should lead to some equations."

Let me try something like that, although not aiming to get equations of motion.

If by hypothesis S = ∫Ldt, L=KE × PE, and δS = 0, then δ∫(KE × PE)dt = 0.

So, if the rate of change of either KE or PE is 0, then in such a world either KE or PE is constant. What would the world be like if either KE or PE could not change? Could there be interactions in the sense that we know at all? Change (as we know it) would seem to be impossible in some sense.

Again, if by hypothesis S = ∫Ldt, L=KE/PE, and δS = 0, then δ∫(KE/PE)dt = 0.

So, the rate of change of
PE can't ever be 0 in such a world, and the rate of change of KE must be 0, i.e. KE must be a constant, in that world. What would a world be like in which every system had the same kinetic energy and no system's potential energy ever stopped changing? What I can think of is a world of particles of the same type beginning from an origin, all accelerating away from one another; KE is the same for each, and PE keeps decreasing but never reaches 0.

I don't know. That's what I came up with.

Maybe you're right, though, that the units involved make my inferences invalid. I tried to keep everything as much the same as possible, while seeing just what it might be like in such strange cases.

8. Jan 26, 2016

### Staff: Mentor

Why? A product can stay constant if both factors vary at the same time in the right way.
Same question for the ratio, but there I'm worried about zero potential with an arbitrary potential definition.

9. Jan 26, 2016

Staff Emeritus
Let's consider KE/PE. We want to minimize this, and since KE is positive and PE can be negative, the solution in most cases will be to make KE as large as possible and PE as small and negative as possible. Defining kinetic energy as 1/2 mv^2 and for the -1/r potential (note that we no longer have the ability to add an arbitrary offset to potential energy) and the solution is for the particle to flit away to infinity at infinite speed. Not to give you equations of motion.

10. Jan 27, 2016

### Staff: Mentor

The solution to "minimizing" the integral over KE-PE in a constant potential is not a stopping particle although that would minimize KE-PE (and also KE+PE and KE*PE and KE/PE if PE is positive).

11. Jan 27, 2016

### crastinus

Thanks for the posts. I need to look more carefully at the math.

I got put on to this question when I started thinking about phase space. I wondered whether, on the analogy of a phase space in the ordinary sense we might also consider a space of phase spaces, let's call it a framespace. This framespace is n dimensional. Each dimension represents some kind of theoretical option for physical properties and movement along a given axis means movement through abstract values for that generalized property (or even through things that are more obviously values like the gravitational constant; e.g. a has some G at x, b has a G at x + k, etc.; where a and b are points along the same axis.) A coordinate within the framespace would be a phase space in which the dimensions represent the different fundamental properties of physical objects in that phase space, mass, charge, etc., as well as the tendencies to obey certain equations of motion, etc. (or however one wants to put that point).

Maybe somebody's already thought of this kind of thing, but isn't that interesting?

What sort of function would we use to go from one phase space to another?

It has been proposed that the laws of physics may themselves evolve: I doubt that, but at any rate this idea of a framespace could actually model such a change. In which case, our universe becomes something like a function in a space of phase spaces!