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# The shearing sheet

## Equations of motion to first order

We have is a patch of the disk centered on some $(R_0,\phi_0)$ that is shearing at a rate $\Omega(R)$. We derive the dynamics of stars in this sheet in coordinates $x = R-R_0$ and $y = R_0(\phi-\Omega_0\,t+\phi_0)$. We can always easily define the coordinate system such that $\phi_0 = 0$, so we will assume that from now on. 
To derive the equations of motion, we write down the Lagrangian (see [here](http://astro.utoronto.ca/~bovy/AST1420/notes-2019/chapters/I-02.-Elements-of-Classical-Mechanics.html#Lagrangian-formulation-of-classical-mechanics)). In polar coordinates, the Lagrangian is

\begin{equation}\label{eq-lagrangian-polar}
    \mathcal{L} = \frac{1}{2}\,\left(\dot{R}^2 + R^2\,\dot{\phi}^2\right) - \Phi(R,\phi)\,.
\end{equation}

Writing this in terms of $(x,y)$, we first of all need $(\dot{x},\dot{y})$:

\begin{align}
    \dot{x} & = {\mathrm{d}\over \mathrm{d} t}\,\left(R-R_0\right) = \dot{R}\,,\\
    \dot{y} & = {\mathrm{d}\over \mathrm{d} t}\,\left(R_0\,[\phi-\phi_0]\right) = R_0\,\dot{\phi}- R_0\,\Omega_0\,,
\end{align}

Thus we have that

\begin{align}
    R & = R_0 + x\,,\\
    \phi & = {y + R_0\,\Omega_0 t \over R_0}\,,\\
    \dot{R} & = \dot{x}\,,\\
    \dot{\phi} & = {1 \over R_0}\,\left(\dot{y} +R_0\,\Omega_0\right)
\end{align}

Plugging these into Equation \eqref{eq-lagrangian-polar}, we get the Lagrangian as a function of $(x,y,\dot{x},\dot{y})$

\begin{equation}\label{eq-lagrangian-shearing}
    \mathcal{L} = \frac{1}{2}\,\left(\dot{x}^2 + {(R_0+x)^2 \over R_0^2}\,\left[\dot{y} +R_0\,\Omega_0\right]^2\right) - \Phi\left(R_0+x,{y + R_0\,\Omega_0 t \over R_0}\right)\,.
\end{equation}

The generalized momenta are then

\begin{align}
    p_x & = {\partial \mathcal{L}  \over \partial \dot{x}} = \dot{x}\,,\\
    p_y & = {\partial \mathcal{L}  \over \partial \dot{y}} = {(R_0+x)^2 \over R_0^2}\,\left[\dot{y} +R_0\,\Omega_0\right]\,.
\end{align}

The main purpose of using the shearing sheet for our purposes is that we will assume that both $|x|$ and $|y| \ll R$, such that we can keep terms only to first order in $x$, $y$, $\dot{x}$, and $\dot{y}$. The generalized momenta in this approximation are

\begin{align}
    p_x & = {\partial \mathcal{L}  \over \partial \dot{x}} = \dot{x}\,,\\
    p_y & = {\partial \mathcal{L}  \over \partial \dot{y}} = R_0\,\Omega_0 + \dot{y} + 2\,x\,\Omega_0\,.
\end{align}

Using Lagrange's equations, we can then derive the equations of motion, again only keeping terms up to first order in the small quantities:

\begin{align}\label{eq-shearing-eom-1}
    \ddot{x} & =  R_0\,\Omega_0^2 +2\,\dot{y}\,\Omega_0 + x\,\Omega_0^2 - {\partial \Phi \over \partial x}\,,\\
    \ddot{y} + 2\,\dot{x}\,\Omega_0 & = - {\partial \Phi \over \partial y}\,.
\end{align}

We will write the gravitational potential as the sum of an axisymmetric contribution $\Phi_\mathrm{axi} \equiv \Phi_\mathrm{axi}(R) = \Phi_\mathrm{axi}(x)$ and a perturbation $\tilde{\Phi}$ to this: $\Phi = \Phi_\mathrm{axi}+ \tilde{\Phi}$. For the axisymmetric part, we have that

\begin{align} 
    {\partial \Phi_\mathrm{axi} \over \partial x} & =     {\partial \Phi_\mathrm{axi} \over \partial R} \\
    & = R\,\Omega^2\\
    & = (R_0+x)\,(\Omega_0 + \Omega_0' x)^2\\
    & \approx R_0\,\Omega_0^2 + 2\,R_0\,\Omega_0\,\Omega_0'\,x + x\,\Omega_0^2\,,
\end{align}

where $\Omega \approx \Omega_0 + \Omega_0'\,x$ for small $|x|$ and $\Omega_0' = \mathrm{d} \Omega / \mathrm{d} R$ evaluated at $R_0$. Plugging this into Equation \eqref{eq-shearing-eom-1}, we find that

\begin{align}
    \ddot{x} & =  2\,\dot{y}\,\Omega_0 - 2\,R_0\,\Omega_0\,\Omega_0'\,x -{\partial \tilde{\Phi} \over \partial x}\,.
\end{align}

Using the Oort constant $A = -R\,\mathrm{d} \Omega /\mathrm{d} \mathrm{R} / 2$ evaluated at $R_0$, we can write the full set of equations of motion as

\begin{align}
    \ddot{x} -2\,\Omega_0\,\dot{y} - 4\,A\,\Omega_0\,x & = -{\partial \tilde{\Phi} \over \partial x}\,,\\
    \ddot{y} + 2\,\Omega_0\,\dot{x} & = - {\partial \tilde{\Phi} \over \partial y}\,,
\end{align}

which are the basic equations of motion in the shearing sheet.

## Equations of motion to second order

To get a more precise solution, we can use the equations of motion that keep terms up to second order in the small quantities. The generalized momenta are then

\begin{align}
    p_x & = \dot{x}\,,\\
    p_y & = R_0\,\Omega_0 + \dot{y} + 2\,x\,\Omega_0 + x^2\,{\Omega_0 \over R_0} + 2\,{x\,\dot{y} \over R_0}\,
\end{align}

Using Lagrange's equations, we again derive the equations of motion, but now keeping terms to second order

\begin{align}\label{eq-shearing-eom-2nd-1}
    \ddot{x} = R_0\,\Omega_0^2 + 2\,\dot{y}\,\Omega_0 + x\,\Omega_0^2 + 2\,{x\over R_0}\,\dot{y}\,\Omega_0+{\dot{y}^2\over R_0} & - {\partial \Phi \over \partial x}\,,\\
    \left[1+2\,{x \over R_0}\right]\,\ddot{y} + 2\,\left[1+{x\over R_0}\right]\,\dot{x}\,\Omega_0  + 2\,{\dot{x}\,\dot{y} \over R_0} & = - {\partial \Phi \over \partial y}\,.\label{eq-shearing-eom-2nd-2}
\end{align}

We can again split the potential into an axisymmetric and non-axisymmetric part and write the axisymmetric derivative as

\begin{align} 
    {\partial \Phi_\mathrm{axi} \over \partial x} & =     {\partial \Phi_\mathrm{axi} \over \partial R} \\
    & = R\,\Omega^2\\
    & = (R_0+x)\,(\Omega_0 + \Omega_0' x + \Omega_0''\,x^2/2)^2\\
    & \approx R_0\,\Omega_0^2 + 2\,R_0\,\Omega_0\,\Omega_0'\,x + x\,\Omega_0^2 + 2\,x^2\,\Omega_0\,\Omega_0' + R_0\,\Omega_0\,\Omega_0''\,x^2+R_0\,\Omega_0'^2 x^2\,,\label{eq-axipot-2nd}
\end{align}

where $\Omega \approx \Omega_0 + \Omega_0'\,x + \Omega_0''\,x^2/2$ now for small $|x|$ and $\Omega_0'' = \mathrm{d}^2 \Omega / \mathrm{d}^2 R$ evaluated at $R_0$. Plugging this into Equation \eqref{eq-shearing-eom-2nd-1} and \eqref{eq-shearing-eom-2nd-2}, we find that

\begin{align}
    \ddot{x} -2\,\dot{y}\,\Omega_0 -4\,\left[1+{x\over R_0}\right]\,\Omega_0\,A\,x- 2\,{x\over R_0}\,\dot{y}\,\Omega_0-{\dot{y}^2\over R_0} 
       + R_0\,\Omega_0\,\Omega_0''\,x^2 +4\,A^2\,{x^2\over R_0} & = 
    - {\partial \Phi \over \partial x}\,,\\
    \left[1+2\,{x \over R_0}\right]\,\ddot{y} + 2\,\left[1+{x\over R_0}\right]\,\dot{x}\,\Omega_0  + 2\,{\dot{x}\,\dot{y} \over R_0} & = - {\partial \Phi \over \partial y}\,.
\end{align}


## Full equations of motion

We can also work out the full, non-linear equations of motion. For this we use the full momenta

\begin{align}
    p_x & = {\partial \mathcal{L}  \over \partial \dot{x}} = \dot{x}\,,\\
    p_y & = {\partial \mathcal{L}  \over \partial \dot{y}} = {(R_0+x)^2 \over R_0^2}\,\left[\dot{y} +R_0\,\Omega_0\right]\,.
\end{align}

and their derivatives

\begin{align}
    \dot{p}_x & = \ddot{x}\,,\\
    \dot{p}_y & = 2\,{(R_0+x) \over R_0^2}\,\left[\dot{y} +R_0\,\Omega_0\right]\,\dot{x} + {(R_0+x)^2 \over R_0^2}\,\ddot{y}\,.
\end{align}

Lagrange's equations then give

\begin{align}
    \ddot{x} & = {(R_0+x) \over R_0^2}\,\left[\dot{y} +R_0\,\Omega_0\right]^2 -{\partial  \Phi \over \partial x}\,,\\
    {(R_0+x)^2 \over R_0^2}\,\ddot{y} & = 
    -2\,{(R_0+x) \over R_0^2}\,\left[\dot{y} +R_0\,\Omega_0\right]\,\dot{x}-{\partial  \Phi \over \partial y}\,.
\end{align}

If we now work out the axisymmetric force to second order as in Equation \eqref{eq-axipot-2nd}, we get

\begin{align}
    \ddot{x} & = {(R_0+x) \over R_0^2}\,\left[\dot{y} +R_0\,\Omega_0\right]^2 
    - R_0\,\Omega_0^2 - (\Omega_0^2-4\,\Omega_0\,A)\,x - (R_0\,\Omega_0\,\Omega_0''+4\,{A^2 \over R_0}-{4\,\Omega_0\,A\over R_0} )\, x^2 -{\partial  \tilde{\Phi} \over \partial x}\,,\\
    {(R_0+x)^2 \over R_0^2}\,\ddot{y} & = 
    -2\,{(R_0+x) \over R_0^2}\,\left[\dot{y} +R_0\,\Omega_0\right]\,\dot{x}-{\partial  \tilde{\Phi} \over \partial y}\,.
\end{align}

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