# Why are 2 dimensions needed for every 1 dimension of space in order to determine the motion of a physical system?

In classical mechanics, the phase space of a mechanical system has twice the number of dimensions of "actual" space (i.e. position space). That is, in phase space, each particle has both a position and a velocity (or similarly, momentum), which constitute separate dimensions. So for a single particle moving in 3-dimensional Euclidean space, the corresponding phase space is 6-dimensional. The phase space is important because it is considered to encompass the entire state of the system. It is a postulate that phase space contains all those, and only those, quantities necessary to predict the future motion of the particles. I'm not aware of any more fundamental derivation of this postulate, and my question is, is there one?

I'm not asking for a derivation from Newtonian mechanics of the fact that phase space looks like this. That derivation is clear. It follows from the fact that a 2nd-order differential equation for position can be converted into two 1st-order differential equations, one for position and the other for velocity/momentum. So formulating mechanics in terms of 1st-order differential equations on a phase space (e.g. Hamiltonian mechanics) is equivalent to 2nd-order differential equations on position space (Newtonian mechanics).

What I'm asking is, why does nature look like this? That is, why does calculating any mechanical motion require us to use 2 dimensions for every 1 spatial dimension (or equivalently, why is Newton's 2nd law 2nd-order in the time derivative rather than 1st-order)? I know that's the way things are, as determined by experiment. What I would like to know is why they are like that, namely a more fundamental explanation for this experimental result.

If this is actually how nature behaves, then this means that either (1) (the Hamiltonian picture) the universe is really 6 dimensional, not 3 dimensional, or (2) (the Newtonian picture), the universe is 3 dimensional, but it has a "short-term memory" in which it looks back at previous times to obtain velocity in addition to position, in order to determine its next state. One may argue that (2) isn't the case because velocity is "instantaneous"--but if this is true, then we're back to (1).

I find both of these two equivalent scenarios logically unsatisfying. That is, why do we only see 3 dimensions, if there are actually 6? The other 3 of them are accessible to us only indirectly (by observing velocities, which require some amount of elapsed time to observe). It is as if our universe is doing calculations in the "background", on a separate, invisible world of 3 additional dimensions. Or similarly, it is as if the 3 dimensions we observe are just a shadow (i.e. a projection) of what is "actually happening" in 6 dimensions. The simplest, most straightforward universe would presumably just use exactly what is here, in the exact present, in our 3 visible dimensions of space, to determine what happens next.

Take the example of a mass on a spring undergoing simple harmonic motion in one dimension, starting from rest. In the corresponding 2-dimensional phase space, this motion traces out a circle. But in actual space, this motion is back and forth with sinusoidal displacement. The motion we see is a projection to 1 dimension of the 2-dimensional motion in phase space. Once again, why would we be living in a projection? Why wouldn't all motion just simply coincide with the spatial dimensions we directly observe?

Interesting, the fact that mechanics occurs in phase space, together with a symmetry principle, implies Galilean relativity. Namely, the symmetry of physical laws under translation in the velocity coordinates is exactly Galilean relativity. (And assuming translation symmetry in the position coordinates implies conservation of momentum.) So any explanation for the above question will also hopefully explain why Galilean relativity holds in our universe.

One may try to invoke time-inversion symmetry (symmetry of Newton's 2nd law under $$t \rightarrow -t$$) to imply that Newton's 2nd law should be 2nd-order in the time derivative. But actually, this implication doesn't hold. For example, if all particles happened to come in pairs, with trajectories $$p_1(t)$$ and $$p_2(t)$$ satisfying $$p_1(t) = p_2(-t)$$, then time-inversion would always be satisfied, regardless of the order of the differential equations of motion. But of course, we don't have pairs of particles whose positions are governed by a 1st-order Newtonian equation. We have single particles whose positions are governed by a 2nd-order equation.

Lagrangian mechanics and the principle of least action don't resolve this question either, because both particle positions and velocities are included in the Lagrangian at the outset of the formalism, making the situation again equivalent.

Finally, one may notice that in quantum mechanics, the Schrödinger equation for a particle is 1st-order in time. But this equation is complex-valued, so it is actually an equation for 2 real-valued wave-functions. Thus we are back to a similar situation, with two real 1st-order differential equations for the evolution of one particle moving in one dimension.

So in summary, I am aware of no simple symmetry principle or other argument for why a particle needs two dimensions for every one dimension of space, in order to determine its motion. Is there such a principle? (This is not a philosophical question. I'm asking about the existence of a mathematically precise principle.)

Does anyone know of any literature on this question?

• The reason Hamiltonian dynamics lives in a space twice the size of the physical space is because dynamical laws are expressed in terms of second time derivatives (you said so yourself).... – QuantumDot Apr 27 at 5:10
• ... And this fact traces all the way back to quantum field theory, wherein the kinetic terms are second order in the derivatives. The reason for this, in the modern language of effective field theories, is that Lorentz invariance forbids first derivatives, and the higher order derivatives are suppressed by the UV scale. So in the low energy, non-relativistic, classical limit, the emergent Newton's laws are in terms of second time derivatives. Notice this is completely circumstantial rather than arising from any deeper principle. – QuantumDot Apr 27 at 5:12
• Essentially a duplicate of physics.stackexchange.com/q/18588/2451 and links therein. – Qmechanic Apr 27 at 7:11
• @QuantumDot, thanks. Saying that Lorentz invariance forbids first order time derivatives is, I think, on the right track. I would like to see a more thorough treatment of this. – Cooler Paradox Sep 29 at 21:39
• @Qmechanic thank you for the link. None of the answers in the linked post, except for the first, speak about the exclusion of 1st order derivatives. Regarding higher-order derivatives, most answers seem to cite the Ostrogradsky instability, which is a specific result in Lagrangian mechanics. I am not convinced that it applies to all possibilities for equations of motion. Finally, the second order nature of mechanics is something we actually feel and experience every day. It should presumably have a simple explanation, not an overly technical or esoteric result. Maybe that's wishful thinking. – Cooler Paradox Sep 29 at 22:22