Undergrad On the definition of canonical coordinates in phase space

[cianfa72]

TL;DR

About the definition of canonical coordinates in phase space (cotangent bundle)

I've a doubt regarding the definition of canonical coordinates in phase space.
As far as I can tell, phase space $T^*M$ is the cotangent bundle of the system configuration space $M$.

$M$ is assumed to be a differential manifold with atlas $A=\{ U_i, \phi_i \}$. Call $q_i$ the coordinate functions of $(U_i, \phi_i)$. They define in $\pi^{-1}(U_i)$ the covector basis field $\{dq_i\}$.

Now at each $p \in M$ any element of the cotangent space $T^*_pM$ can be written as linear combination of $\{dq_i\}$. Call $\{ p_i \}$ the set of functions defining the components of cotangent element at any point $p$. Hence any cotangent field (one-form) $\theta$ can be written as
$$\theta = \sum_i p_idq_i$$
Assuming the above is correct, are those $\{ q_i,p_i \}$ the canonical coordinates for phase space (cotangent bundle) ?

 

[martinbn]

Yes.

 

[cianfa72]

Ok, so basically $\{ p_i \}$ are coordinates in any $\pi^{-1}(p) \cong F$ with respect to $\{ dq_i \}_p$ picked as basis for covectors at $T^*_pM$.

 

[martinbn]

Ok, so basically $\{ p_i \}$ are coordinates in any $\pi^{-1}(p) \cong F$ with respect to $\{ dq_i \}_p$ picked as basis for covectors at $T^*_pM$.

Yes, but be careful with the notations. You denote a point in the base manifold $M$ by $p$, but your coordinates on $M$ are $\{q_i\}$, and the $p_i$'s are the components of the one-forms in the basis $\{dq_i\}$.

 

[cianfa72]

Yes, but be careful with the notations. You denote a point in the base manifold $M$ by $p$, but your coordinates on $M$ are $\{q_i\}$, and the $p_i$'s are the components of the one-forms in the basis $\{dq_i\}$.

Ah ok. To avoid confusion let's call $a$ the point in the base manifold $M$, so that $\{ p_i \}_a$'s are the components of the one-forms (covector fields) evaluated at $a$ in the coordinates basis $\{dq_i\}_a$.

Btw, by a diffeomorphism between $\{p_i\}$ and $\{z_i\}$, in the new coordinates one-forms $\theta$ are no longer written in the form $$\theta = \sum_i z_idq_i$$ right ? As far as I can tell, the above expression for $\theta$ holds true only in canonical coordinates.

 

[martinbn]

Btw, by a diffeomorphism between $\{p_i\}$ and $\{z_i\}$, in the new coordinates one-forms $\theta$ are no longer written in the form $$\theta = \sum_i z_idq_i$$ right ? As far as I can tell, the above expression for $\theta$ holds true only in canonical coordinates.

What do you mean by a diffeomorphism? Between what? The contangent bundle and itself or something else? The coordinates of the cotangent bundle are $(q_i,p_i)$.

 

[cianfa72]

What do you mean by a diffeomorphism? Between what? The contangent bundle and itself or something else? The coordinates of the cotangent bundle are $(q_i,p_i)$.

I meant a differentiable one-to-one onto map with inverse differentiable from the $\{p_i\}$ to say the $\{z_i\}$.

A one-form (covector field) $\theta$ written as $\theta = \sum_i p_idq_i$ in $\{ q_i,p_i \}$ canonical coordinates can no longer be written as $\sum_i z_idq_i$ in the new $\{q_i,z_i \}$ cotangent bundle coordinates.

 

[martinbn]

I meant a differentiable one-to-one onto map with inverse differentiable from the $\{p_i\}$ to say the $\{z_i\}$.

My question was about the manifold. A diffeomorphism of which manifold? My guess is the cotangent bundle. But these, that you've written above are not coordinates of the cotangent bundle.

A one-form (covector field) $\theta$ written as $\theta = \sum_i p_idq_i$ in $\{ q_i,p_i \}$ canonical coordinates can no longer be written as $\sum_i z_idq_i$ in the new $\{q_i,z_i \}$ cotangent bundle coordinates.

Why not? What is the relationship between the p's and the z's?

 

[cianfa72]

Sorry, I'm confused a bit. I took a look at Lee - Introduction to Smooth Manifold.

Long story: starting from a $m$-dimensional smooth manifold $M$ and its maximum atlas, one can define a corresponding atlas for the cotangent bundle $T^*M$ turning it into a differentiable manifold on its own. Said $q^i$ the $U$'s chart coordinates for $M$, take as $p_i$ the coordinates that assign to each covector at point $a \in U$ in the fiber its components in the coordinate basis $\{ dq^i \}_a$. By definition $(q^i,p_i)$ are the canonical coordinates for $\pi^{-1}(U)$.

Now a covector field (one-form) on $T^*M$ is a section of $T^*(T^*M)$. The canonical one-form $\theta$ (aka tautological one form) is defined in canonical coordinates as $$\theta = \sum_i p_idq^i$$ Suppose to define a "chart diffeomorphism" $(q_i,p_i) \mapsto (q_i,z_i)$ that leaves untouched the $q_i$ while the $z_i$ depending only from the $p_i$ (i.e. the diffeomorphism's matrix representation in those canonical coordinates is block diagonal). As far as I can tell, this time the canonical one-form $\theta$ is given in the form
$$\theta = \sum_i f_i(z_1, z_2 ... z_m)dq^i$$ and cannot be written as $\theta = \sum_i z_idq^i$.

 

[martinbn]

Oh, i see, yes you are right. I thought you meant a general one form and a map that only chages the p's. For example if $p=z^2$ and q stays the same, then $pdq$ becomes $z^2dq$. But now i see what you meant.