Why are position and velocity independent variables in a phase space?

[Happiness]

Why do we take a particle's position $x$ and its velocity $\dot{x}$ as independent variables in a phase space when they are dependent in the sense that given the function $x(t)$, we can get the function $\dot{x}(t)$?

I'm thinking they are independent variables but not independent functions. Does this make sense?

 

[Lucas SV]

I would say the variables are independent by assumption. $x$ and $v$ are both defined to be functions of time, and the equation $\dot{x}=v$ becomes a constraint that must hold in order for $v$ to correspond to the physical velocity when $x$ is interpreted as the position of a particle.

 

[Happiness]

I would say the variables are independent by assumption. $x$ and $v$ are both defined to be functions of time, and the equation $\dot{x}=v$ becomes a constraint that must hold in order for $v$ to correspond to the physical velocity when $x$ is interpreted as the position of a particle.

If we drop the assumption that $x$ and $v$ are independent, would we run into any problems?

Say, suppose we define the variable $v$ as $v=2x$. We then plot the $x$-axis as the horizontal axis and the $v$-axis as the vertical axis. Is there anything wrong with plotting two dependent variables as two perpendicular axis?

We say $x$ and $v$ are independent, is it because $v=2x$ is true only for a particle and may not be true for another particle? In this case, is there a mathematical definition for independence, say, in the form of an equation, similar to the one below for the definition of linear dependence of 2 vectors?

 

[Lucas SV]

If we drop the assumption that $x$ and $v$ are independent, would we run into any problems?

Say, suppose we define the variable $v$ as $v=2x$. We then plot the $x$-axis as the horizontal axis and the $v$-axis as the vertical axis. Is there anything wrong with plotting two dependent variables as two perpendicular axis?

We say $x$ and $v$ are independent, is it because $v=2x$ is true only for a particle and may not be true for another particle? In this case, is there a mathematical definition for independence, say, in the form of an equation, similar to the one below for the definition of linear dependence of 2 vectors?

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In the usual particle mechanics phase space is defined as a vector space. More precisely the phase space of a classical particle in 3 dimensionals is a 6 dimensional vector space. This is because the state of a particle at time $t$ is described by 3 position and 3 velocity components. In general phase space is the space of states or configurations of the system.

However a space is not sufficient to describe physics. One needs laws and interpretations. One interprets the motion of a particle as a path in phase space parametrized by time, where each point in the path is the state at a certain time. The laws governing the system are Hamilton's equations. They will tell you which path is the actual path of the particle in phase space.

It is true that the axis for position and momentum are orthonormal by definition. So yes, you can think of the 'position' unit vector and 'momentum' unit vector in phase space as linearly independent in the sense of linear algebra.

 

[Happiness]

In the usual particle mechanics phase space is defined as a vector space. More precisely the phase space of a classical particle in 3 dimensionals is a 6 dimensional vector space. This is because the state of a particle at time $t$ is described by 3 position and 3 velocity components. In general phase space is the space of states or configurations of the system.

However a space is not sufficient to describe physics. One needs laws and interpretations. One interprets the motion of a particle as a path in phase space parametrized by time, where each point in the path is the state at a certain time. The laws governing the system are Hamilton's equations. They will tell you which path is the actual path of the particle in phase space.

It is true that the axis for position and momentum are orthonormal by definition. So yes, you can think of the 'position' unit vector and 'momentum' unit vector in phase space as linearly independent in the sense of linear algebra.

So can I say that the position unit vector and the momentum unit vector in a phase space are linearly independent, but the position function $x(t)$ of a particle and the velocity function $v(t)$ of a particle are not linearly independent? And that it is nonsensical/ambiguous to say position $x$ and velocity $v$ are dependent or independent?

 

[Lucas SV]

So can I say that the position unit vector and the momentum unit vector in a phase space are linearly independent, but the position function $x(t)$ of a particle and the velocity function $v(t)$ of a particle are not linearly independent? And that it is nonsensical/ambiguous to say position $x$ and velocity $v$ are dependent or independent?

Oh, I said position and momentum out of practise, since these are the fundamental variables in quantum mechanics. In classical physics they are prety much the same except for the mass factor. You can replace momentum by velocity in everything I said. Phase space is defined as the 6 dimensional vector space which contains the position and velocity unit vectors as an orthonormal basis. So in phase space position and velocity are linearly independent (the momentum-position and velocity-position phase spaces are equivalent by scaling).

 

[Happiness]

Oh, I said position and momentum out of practise, since these are the fundamental variables in quantum mechanics. In classical physics they are prety much the same except for the mass factor. You can replace momentum by velocity in everything I said. Phase space is defined as the 6 dimensional vector space which contains the position and velocity unit vectors as an orthonormal basis. So in phase space position and velocity are linearly independent (the momentum-position and velocity-position phase spaces are equivalent by scaling).

Alright, what I get from your replies is that it makes no sense to just say $x$ and $v$ are dependent or independent. Because $\hat{x}$ unit vector and $\hat{v}$ unit vector in phase space are linearly independent, whereas $x(t)$ function and $v(t)$ function are dependent. So technically, we have to be clear and say $\hat{x}$ unit vector or $x(t)$ function. Am I right?

 

[Dale]

Why do we take a particle's position $x$ and its velocity $\dot{x}$ as independent variables in a phase space when they are dependent in the sense that given the function $x(t)$, we can get the function $\dot{x}(t)$?

But we are not given $x(t)$. Otherwise we wouldn't be using phase space to solve for it. For each system there are states which go through each $x$ at any $\dot x$, so they are independent. Then we solve for $x(t)$

 

[Happiness]

But we are not given $x(t)$. Otherwise we wouldn't be using phase space to solve for it. For each system there are states which go through each $x$ at any $\dot x$, so they are independent. Then we solve for $x(t)$

Could we make this explanation more formal? My attempt is as follows:

For each system there are states that go through each $x$ value at any $\dot x$ value, so the variables are independent, but the functions $x(t)$ and $\dot x(t)$ are dependent.

A variable can take on any value in a given set, whereas a function cannot. A function is a relation that maps a value in a given set to another value (which could be distinct or identical) in another given set. (Another is not a good word, but I can't think of a better word.)

Definition of independent variables
Two random variables X and Y are independent if and only if for every a and b, the events {X ≤ a} and {Y ≤ b} are independent events.

Definition of independent events

Definition of independent functions
I can't find it. I can only find the definition of linearly independent functions. But the functions $x(t)$ and $\dot x(t)$ could be linearly independent but dependent.

 

[DrClaude]

Could we make this explanation more formal? My attempt is as follows:

For each system there are states that go through each $x$ value at any $\dot x$ value, so the variables are independent, but the functions $x(t)$ and $\dot x(t)$ are dependent.

I think that the correct way to go about this is to say that $x(t)$ and $\dot x(t)$ are governed by a set of coupled differential equations.

 

[Lucas SV]

Don't confuse phase space with a particle trajectory. The unit vectors I was talking about do not represent a particle. They are linearly independent. But the particle's actual position and actual momentum are not, in general. Unfortunately in physics, we use $x$ to mean both the particle trajectory and the coordinate in phase space/real space. So I will use a different notation for clarity's sake.

A phase space (for a classical 1 particle system in 3 dimensions) is $P=\mathbb{R}^6$. We denote an element of the phase space by $(x_1,x_2,x_3,v_1,v_2,v_3)$. The unit vectors I mentioned previously are nothing but
$$(1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1)$$
The classical trajectory of a particle is a smooth map $\gamma:\mathbb{R} \rightarrow P$ such that $\dot{\gamma}_1=\gamma_4$, $\dot{\gamma}_2=\gamma_5$ and $\dot{\gamma}_3=\gamma_6$, where $\dot{\gamma}=(\dot{\gamma}_1,\dot{\gamma}_2,\dot{\gamma}_3,\dot{\gamma}_4,\dot{\gamma}_5,\dot{\gamma}_6)$ is the derivative of $\gamma=(\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5,\gamma_6)$ and $\gamma_i:\mathbb{R}\rightarrow\mathbb{R}$, $i=1,2,3,4,5,6$. There is a unique particle trajectory $\gamma$ satisfying the initial condition $\gamma(t_0)=u$, for some $u\in P$ and $t_0\in\mathbb{R}$.

The right material for you to learn this from is Dynamical Systems/Classical Physics, in particular the Hamiltonian formalism.

 

[Happiness]

In the derivation of the Euler-Lagrange equation from the principle of stationary action, Morin, in his book, wrote that nowhere do we assume that $x$ and $\dot x$ are independent variables. But in phase space, they are independent variables.

Am I right to say that he should have said "nowhere do we assume that $x$ and $\dot x$ are independent functions" instead? This is because in (5.16), we differentiate $x$ and $\dot x$, and variables cannot be differentiated. For example, we can differentiate a function $y(x)$ wrt $x$, but we can't differentiate a variable $x$ wrt $x$ or a variable $y$ wrt $x$. (If we do $\frac{dx}{dx}$, we are differentiating the function $y(x)=x$ wrt $x$.)

 

[Lucas SV]

In the lagrangian formulation we do not use phase space. The word 'variable' is not used precisely in mathematics, in my experience. You do not need to talk about the variable if you want to talk about the funtion. Learning some logic, set theory and functions in a more systematic way may help you.

I think you are getting too bogged down on this dependent\independent issue. It doesn't matter if you want to understand things precisely, or not so much so, you will have to work hard anyway.

Do some exercises in the lagrangian formulation, and you will understand how to work with it. After you know the lagrangian formulation well, move on to the hamiltonian formulation. Don't worry about learning things too quickly.

When it comes to functionals (the action is a functional), the proper mathamatics can get quite technical. Also if you are rigid about notation you may have problems. Even mathematicians in differential geometry will use many different notations quite flexibly. Do not worry too much about the technicalities and details for now, learn the subject by doing exercises, and later you will have the opportunity to come back to this once you have more experience.

 

[Khashishi]

x is frequently used to mean two related but different things.
The first is a coordinate. There is a set of coordinates for each point in space, or of phase space.
The second is the actual position of an object. This only occupies a curve in space much smaller than the whole space.

The Lagrangian uses the first meaning.

 

[Happiness]

x is frequently used to mean two related but different things.
The first is a coordinate. There is a set of coordinates for each point in space, or of phase space.
The second is the actual position of an object. This only occupies a curve in space much smaller than the whole space.

The Lagrangian uses the first meaning.

How do you reconcile Morin's comment that $x$ and $\dot x$ are dependent variables with the fact that they are independent variables in phase space?

 

[Khashishi]

I'm not sure I agree with Morin. 5.17 and 5.16 are fine even if they are independent.

Euler-Lagrange equation is an optimization problem, so the solution is more constrained than the functional that is being optimized. So $x$ and $\dot{x}$ are independent in the functional, but not in the solution. I think it's analogous to this simpler example. Consider the line
y=mx+b
Clearly, x and y are not independent on the line. Either y is function of x, or x is a function of y.
But now consider the function
f(x,y) = mx+b-y
Now consider the optimization problem: f(x,y) = 0. Obviously the solution is y=mx+b.
But, the function f(x,y) is defined on all x and y. x and y are independent parameters to the function f. But the solution is a constraint between them.

 

[Happiness]

I'm not sure I agree with Morin. 5.17 and 5.16 are fine even if they are independent.

Euler-Lagrange equation is an optimization problem, so the solution is more constrained than the functional that is being optimized. So $x$ and $\dot{x}$ are independent in the functional, but not in the solution. I think it's analogous to this simpler example. Consider the line
y=mx+b
Clearly, x and y are not independent on the line. Either y is function of x, or x is a function of y.
But now consider the function
f(x,y) = mx+b-y
Now consider the optimization problem: f(x,y) = 0. Obviously the solution is y=mx+b.
But, the function f(x,y) is defined on all x and y. x and y are independent parameters to the function f. But the solution is a constraint between them.

Can I say there are two ways of viewing (5.16), depending on the vector space you use to visualize the optimization?

If we use the position-time vector space, then we vary the trajectory $x(t)$ of the particle. In this case, $\dot x(t)$ changes accordingly as $x(t)$ is varied. Then the $x(t)$ and $\dot x(t)$ in (5.16) are NOT independent.

On the other hand, if we use the velocity-position phase space to visualize the optimization, then we vary the phase-space "trajectory" $\begin{pmatrix}x(t)\\\dot x(t)\end{pmatrix}$ of the particle. In this case, $x(t)$ and $\dot x(t)$ are varied independently. Then the $x(t)$ and $\dot x(t)$ in (5.16) are independent.

However, $\frac{\partial x}{\partial a}$ and $\frac{\partial\dot x}{\partial a}$ are related by (5.17) as the latter is the derivative of the former. Does this imply that not all variations of the phase-space "trajectory" $\begin{pmatrix}x(t)\\\dot x(t)\end{pmatrix}$ are considered in the optimization? But only those variations that satisfy (5.17)? If so, then $x(t)$ and $\dot x(t)$ are NOT varied independently. And the function $x(t)$ and the function $\dot x(t)$ in (5.16) are NOT independent.

Also, consider the term $\frac{\partial L}{\partial x}\frac{\partial x}{\partial a}$ in (5.16). Can I say that the $x$ in $\frac{\partial L}{\partial x}$ and the $x$ in $\frac{\partial x}{\partial a}$ mean slightly different things, though they are both symbolized in the same way? The $x$ in the former is a variable (which can take on certain values), whereas the $x$ in the latter is a function (which is a map).

 

[PeroK]

How do you reconcile Morin's comment that $x$ and $\dot x$ are dependent variables with the fact that they are independent variables in phase space?

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As mentioned in post #11, you are confusing a specific particle trajectory with all the possible states of a particle.

in a sense, its no different from x and y being independent variables, in terms of the x-y plane, but not if you looking at a specific graph, like $y = x^2$.

 

[Sam Robertson]

They are only independent in the sense that you need position and velocity to specify the initial state of a system (and thus the further time evolution of the system - you are completely right about this) because F = m a is a 2nd order differential equation and there is an initial velocity integration constant. If the law were different say F = m v you could specify all the initial states with just position because v = dx / dt would just be F(x) / m and there is no initial velocity freedom. Leonard Susskind talks about this at his first lecture at stanford (google) if I am not misunderstanding his lecture.

 

[Sam Robertson]

What I mean is that they are only independent in the sense that you are free to specify them *independently* as initial conditions. Once you do that all you have to do is take some nice derivatives (no integration constants needed!) and you will have the evolution of your system for all time.

 

[Stephen Tashi]

Definition of independent variables
Two random variables X and Y are independent if and only if for every a and b, the events {X ≤ a} and {Y ≤ b} are independent events.

The mathematical definition of "independent events" arises in probability theory, so "independent events" is not the terminology you want to use for a formal mathematical definition of "independent" in a deterministic settling - like classical mechanics.

Definition of independent functions
I can't find it. I can only find the definition of linearly independent functions. But the functions $x(t)$ and $\dot x(t)$ could be linearly independent but dependent.

Yes, it is difficult to give a mathematical definition that expresses the common language meaning of "independent" because the common language meaning of "independent" involves metaphysical questions - such as "What does it mean for one thing to cause or affect another ?"

For example, suppose we define a function $f(x,y) = \sqrt{(x + y)}$ so its domain is the set of points on the plane $\{(x,y):x+y \ge 0\}$. In mathematical terminology, the variables $(x,y)$ are "the independent variables". However, in terms of common language notions, if we pick a value of $x$ then this "affects" what choices we have for $y$.

Aside from "linearly independent", there are various informal uses of the work "independent" in mathematics, but I don't know of any formal definition that encompasses all situations. We could try the definition "$A$ is independent of $B$ means that $A$ is not a function of $B$", but that doesn't completely capture the common language notion of "$A$ does not affect $B$".

The clearest way to resolve your original post is to admit that physics uses terminology that is atrocious, from the point of view of pure mathematics. In mathematics, a "function" is defined so it has a (fixed) domain and co-domain. If you talk about a function f with one domain and co-domain in the first paragraph on a page and then begin to talk about a function f with a different domain and co-domain on the bottom of the page then you are talking about two different functions , so you shouldn't name them both "f".

For example, if the first paragraph says: "Define f(x,y) = 3x + 2y" and the last paragraph says "x = 4t and y = 2t, so f(t) = 12t + 4t" this is self-contradictory terminology from a purely mathematical point of view because "f" in the first paragraph has a different domain than "f" in the last paragraph. In the first paragraph an element of the domain is a pair of real numbers (x,y). In the last paragraph, an element of the domain is a real number t.

In physics (and applied mathematics) what is denoted as "the function f", often doesn't refer to a mathematical function. Instead "f" refers to some phenomena like "the position of the particle", "the cost of tea", "the length of the cable" etc. If we think of "f" as phenomena then as we reason about "f" in various ways, we can keep changing and modifying how we model "f" as a mathematical function. So we use the same name "f" for a collection of different functions.

If we think of "position" and "velocity" as phenomena associated with the same particle then we might be able to express "velocity" as a function of position. If the particle moves in a figure eight pattern then there is a position where it has two different velocities . In that case, velocity cannot be expressed as a function of position. However, there are cases where velocity can be expressed a function of position - such as the typical "falling body" problem.

To me, the most confusing aspects of physics texts involve the question of "what is a function of what" in the context of derivatives and partial derivatives. Suppose we read that $L$ is defined as a function of $(x,v)$. If the expression $\frac{dL}{dt}$ appears in a later paragraph then this only makes sense (mathematically) if $L$ is defined as a function of the single variable $t$. The name "$L$" is being used to denote both a function of $t$ and a mathematically different function of $(x,v)$. There are further complications if the text says that "$v(t) = \frac{dx}{dt}$" because the name "$x$" is being used for both an element in the domain of a function and "$x$" is also being used for the name of a function (i.e. $x(t)$ denotes something in the co-domain of the function whose name is "$x$".)

To interpret a physics text, you must keep in mind that a phenomena like "velocity" or "the cost of tea" can be modeled by several different mathematical functions and that the text will often use the same name for the phenomena itself and also for each of the several different functions that model it.

For concepts involving phase spaces, you have to unscramble whether the name for a phenomena like "velocity" refers to a very specific phenomena (e.g. the "velocity" of a particular particle on a particular trajectory") or whether "velocity" refers to an element in set of more general phenomena (e.g. some velocity chosen from the set of velocities belonging to all possible particles on all possible trajectories).

 

[Zai-Qun YU]

Actually, I would like to highlight the meaning of "state function". It is a state at a certain time point. At any point of time, one particle can be described by six variables, three for position and three for velocity. A particle in position (x0,y0,z0) can move with a range of velocity, such as (Vx1, Vy1, Vz1), or (Vx2,Vy2,Vz2), etc. Position and velocity are independent in the sense that the state of a particle, or the system state, is not defined without either of these two.