# What exactly is Phase space

#### aniket7tomar

What exactly is phase space and how is it different from real space i.e. 3 co-ordinate system?
what does it mean when someone says "dynamics occurs in phase space"?
I'm very new to all this so pls take that into account to

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#### DrClaude

Mentor
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#### aniket7tomar

Yes, i did but i could not get a feel of it...i mean i'm not able to put it in perspective with real world the way i'm able to do with 3 co-ord. system......
I also tried looking at the "space" page of wikipedia but could not understand it either...
Tnx

#### haael

Yes, i did but i could not get a feel of it...i mean i'm not able to put it in perspective with real world the way i'm able to do with 3 co-ord. system......
Drop this intuition completely. Do not compare the phase space to the physical space. They are completely different concepts.

Phase space is a mathematical tool. It's a set of all possible system configurations. You can say, the set of all possible universes. A point in this space is a one particular universe at one moment of time. The metric in the phase space says how similar two universes are.

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#### Pythagorean

Gold Member
The phase space is the complete set of variables in a system of differential equations. So if you have 2 differential equations (with two variables total) then you can plot state trajectories on paper using the xy axes. If your system is 3D, then you need x, y, and z axes to represent the phasespace of the system. Its difficult to imagine a 4D system (but still doable with some cleverness).

They are not necessarily spatial variables, but they can be. The point is to analogize the variable space to real space to help our intuition to see how the system behaves. So we can imagine the system as it changes state as a particle moving through space.

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#### DrClaude

Mentor
Have a look at the Wikipedia page on simple harmonic motion. You will get a feel for what a trajectory in phase space is.

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#### DrClaude

Mentor
okay, now i understand it somewhat...but can you also please tell me what the proffesor in the video means whenhe says "dynamics is not happening in real space but in phase space which is both velocity and co-ordinates" at 19:45
The coordinates of the particles are not sufficient to describe the dynamics of the system, you need also the velocities. In that sense, knowing where the particles are in space doesn't tell you how the system will evolve dynamically. On the other hand, knowing both the position and velocity of the particles, which you could call their coordinates (or "position") in phase space uniquely determines the dynamics of the system. The dynamics of the system is a specific trajectory in phase space.

(Note that this is in a context where all the internal and external forces on the system are known and accounted for.)

#### bp_psy

okay, now i understand it somewhat...but can you also please tell me what the proffesor in the video means whenhe says "dynamics is not happening in real space but in phase space which is both velocity and co-ordinates" at 19:45
He means what he was saying for the last couple of minutes before 19:45. The dynamics of a system are not fully described by the coordinates. Having a position at a given time (x(t_o),y(t_o),z(t_0)) is not enough to determine the position immediately after t_o.You also need momentum
(Px(t_o),Py(t_o),Pz(t_0)). This means that the dynamics happen in a 6D phase space for a single free particle.

#### Pythagorean

Gold Member
The coordinates of the particles are not sufficient to describe the dynamics of the system, you need also the velocities. In that sense, knowing where the particles are in space doesn't tell you how the system will evolve dynamically. On the other hand, knowing both the position and velocity of the particles, which you could call their coordinates (or "position") in phase space uniquely determines the dynamics of the system. The dynamics of the system is a specific trajectory in phase space.

(Note that this is in a context where all the internal and external forces on the system are known and accounted for.)
For example, if a pendulum is swinging, it passes through theta=0 (straight up and down). But passing through theta=0 from left to right is a different state than passing through theta=0 from right to left. In each case, the velocities are different, so the velocity AND the position are both required to give a full description of the system's state.

#### aniket7tomar

Okay, so we say dynamics doesn't occur in real space because all the different states can't be determined for which we'll also need velocities and in that sense physics of stationary objects or statics can occur in real space....
Also saying dynamics doesn't occur in real space and saying that the system concerned is changing positions in real world are both true and not contradictory because they are two completely different things.......
Pls tell me if i'm wrong somewhere and tnx for helping out

#### Chestermiller

Mentor
If you were solving for the motion of n interacting particules using Newton's second law, you would be solving 6n equations in 6n unknowns: 3 spatial coordinates and 3 velocity components for each particle. The 6n is often referred to as the number of "degrees of freedom" of the system. The 6n unknowns are referred to as the phase space variables.

Chet