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I am new in classical physics and I frequently come across the terms phase space and phase trajectory. Can anyone please explain to me what they are in a simple language?

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    $\begingroup$ wiki article $\endgroup$ Oct 17, 2017 at 12:47
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    $\begingroup$ Actually I did read the Wikipedia article. But as I couldn't understand a thing there, I posted the question here. If possible, please explain it to me in a simple language $\endgroup$
    – user214671
    Oct 17, 2017 at 12:49

3 Answers 3

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It is not precise what you mean by simple language, but let's try a very basic approach. Bare in mind that phase space is an abstract concept which is based on previous abstractions of space and system, with which you need to be familiar.

  • space is used in the geometrical sense. In classical mechanics mainly referred to some Euclidean space. It can be defined as the set of all positions (or points) that can be expressed by providing a pair (2 dimensions) or a trio (three dimensions) of Real numbers. A given combination of these numbers defines a unique position in this space. The amount of numbers needed corresponds to the amount of dimensions of this space, hence you can imagine a space of 4, or 5, or even an infinite number of dimensions.

  • system in classical mechanics usually refers to a point mass or a group of point masses that can move in a plane (2 dimensions) or in space (3 dimension). This system can be described by listing the coordinates of each of the point masses and their velocities. The number of spatial dimensions (space in which they exist and move) defines the amount of numbers needed to describe the system: 2 spatial dimensions implies that two numbers for the position of each point mass and two numbers for their velocities are necessary. Usually the time is a necessary number, that tells you what moment of the system you are referring to.

  • phase space is the space formed by all the points that describe the phases or states of the system. In other words, is a way of representing different forms or configurations that our system can have. This space is made by all the possible combinations of numbers describing the system. If the system has one point mass moving in a line, you need two numbers: the x coordinate and the velocity along this line. This system can be represented in a plane (a 2 dimensional Euclidean). If the system is N mass points moving all in the same line, you need 2 values for each of the masses, 2N values. This space is hard to represent because we only can imagine up to three dimensions in which we live. If the system has N point masses moving all in the same spatial plane, you need 4 numbers per each point mass: 2 for the position ($x$,$y$) and two for the velocities in each spatial component ($v_x$, $v_y$) and in total it would be 4N dimensions for the phase space. Time is not necessary here because in this description the interest is in answering 'how many possible states my system can be in?' instead of 'when can my system be in a given state?'.

For example, if our system is a mass connected to a spring, it can be described as only one point mass with a given position and a velocity along a line. Because we only need these two quantities (position and velocity) the system can be represented in a plane, choosing the velocity values as the $y$ coordinate and the position values as the $x$ coordinate the phase space would look like in the figure. All points are possible values in theory, but the points in blue in the line, represent a specific spring of defined total energy. This is useful because it tells you some information already about the system.

Phase space of a mass spring system

Finally, the phase space can be made of dimensions other than positions and velocities. It is very common to describe the system in terms of positions and momenta, or include spins, etc. The common feature is finding what are the minimal necessary quantities necessary to describe the system, and those will be the dimensions of the phase space.

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The "phase space" is the space composed of position and momentum that characterizes the system.

Imagine a particle in three-dimensional space. Imagine that you set up a cartesian coordinate system $x,y,z$. Then at any instant of time, the position of the particle will be descriple by a triple of numbers $(x,y,z)$. This carries the information of the configuration of the system, where the particle is. The space of all such triples is called the configuration space, denoted usually by $Q$.

But just the position doesn't completely characterizes the state of the particle. There is also the state of motion. The state of motion is well described by the momentum. The momentum is a vector, namely, a triple of numbers $(p_x,p_y,p_z)$ which for a particle in three dimensions with mass $m$ is defined as

$$p_x = mv_x,\quad p_y = mv_y, \quad p_z = mv_z$$

where $(v_x,v_y,v_z)$ is the velocity vector of the particle.

In vector notation we have $\mathbf{p} = m\mathbf{v}$. This $\mathbf{p}$ is a measure of the "quantity of motion" associated to the particle when it has velocity $\mathbf{v}$. Also in vector notation the position $(x,y,z)$ can be written as $\mathbf{r}$, so that we have:

$$\mathbf{r} = (x,y,z),\quad \mathbf{p}=(p_x,p_y,p_z).$$

Now with time fixed, the particle is somewhere with some momentum, does it is described by a pair $(\mathbf{r},\mathbf{p})$. All these pairs form the phase space $M$. These are actually tuples of $6$ numbers $(x,y,z,p_x,p_y,p_z)$.

It turns out that as time passes the position varies im time. This can be described by an association $t\mapsto \mathbf{r}(t)$, where now $\mathbf{r}(t)$ is the position triple at time $t$. The same happens for momentum, as time passes momentum changes and this is described by an association $t\mapsto \mathbf{p}(t)$. This in turns gives an association $t\mapsto (\mathbf{r}(t),\mathbf{p}(t))$.

This is by definition a sequence of points of phase space parametrized by time. This is a phase space trajectory and it characterizes the evolution in time of the particle. This is what is usualy sought in Classical Mechanics.

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  • $\begingroup$ Yet it is perfectly possible to have the sphere $S^2$ as a phase space, with Poisson bracket $\{f,g\}=\frac{1}{\sin\theta}\left( \frac{\partial f}{\partial \varphi}\frac{\partial g}{\partial \theta} - \frac{\partial f}{\partial \theta}\frac{\partial g}{\partial \varphi}\right) $, for which $\varphi$ plays the role of position and $\cos\theta$ the role of momentum, the latter unrelated to "quantity of motion". $\endgroup$ Sep 4, 2018 at 14:33
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The phase space is a diagram in which the horizontal axis is position $x$, and the vertical axis is momentum $p$.

You know that momentum is defined as $mv$ in classical physics, so it is just proportional to velocity. IF you want, "it's the velocity scaled by the mass". Of course it has much more meaning behind that, but it can be useful to see it like that for those diagrams. It's LIKE representing velocity↔position.

I'm considering movement in only 1D, since more dimensions would require more axis and that's not plottable in 2D.

So that's the face space. Trajectories in that space represent pairs of $(x, p)$ of the particle's movement.

An easy example is an oscillation from $x=-A$ to $x=A$.

First the position is $-A$ and the velocity is locally 0 but it soon starts moving to $+A$, so it is positive. Then, when it reaches the equilibrium (x=0) the velocity is máximum. Now the velocity starts decreasing as it approaches to +A.

You can see that this oscillation in 1D can be seen as a clockwise circle in the phase space.

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