I am given the equations of motion: $$\dot{r} = f(r)p_r,$$ $$\dot{p_r} = -\frac{V'(r)}{2f(r)}-\frac{f'(r){p_r}^2}{2}+\frac{V(r)-E^2}{2f^2(r)}f'(r)$$ Along with the conditions that $$E^2-V(r_0)=V'(r_0)=0$$

I need to linearize the equations of motion around a circular orbit at $r=r_0$ so that I get: $$\delta\dot{r}=\delta p_r,$$ $$\delta \dot{p_r}=-\frac{V''(r_0)\delta r}{2f(r_0)}$$

How to get the solution by using the concept of linearization?


I tried to calculate the Jacobian matrix using Mathematica:

jacobianMatrix = 
 Grad[{f[r] p[r], -V'[r]/(2 f[r]) - (f'[r] p[r]^2)/
        2 + (V[r] - En^2)/(2 f[r]^2) f'[r]}, {r, p[r]}] /. r -> r0 /. 
    En^2 -> V[r0] /. V'[r0] -> 0 // MatrixForm

which gives me the output:

$$\left( \begin{array}{cc} p_{r_0} f'({r_0}) & f({r_0}) \\ -\frac{1}{2} p_{r_0}^2 f''({r_0})-\frac{V''({r_0})}{2 f({r_0})} & -p_{r_0} f'({r_0}) \\ \end{array} \right)$$

where I am getting the $-\frac{V''(r_0)}{2f(r_0)}$ term along with extra terms, I am not sure why the other terms vanish.

Source: Probing phase structure of black holes with Lyapunov exponents (page 13)

  • $\begingroup$ What do you know about linearization, what is your approach to it? It can be systematic using partial derivatives or procedural/manual by inserting linear Taylor polynomials, or dual numbers if this is more approachable to you. $\endgroup$ Commented May 15, 2023 at 8:15
  • $\begingroup$ @LutzLehmann I tried doing it using partial derivatives but got stuck. $\endgroup$
    – codebpr
    Commented May 15, 2023 at 8:35
  • 2
    $\begingroup$ Then please document how and where you got stuck. Add this to the question text. $\endgroup$ Commented May 15, 2023 at 8:53
  • $\begingroup$ @LutzLehmann I have shown the details in the edit. $\endgroup$
    – codebpr
    Commented May 16, 2023 at 7:31
  • $\begingroup$ The radial impulse component is zero, for a circular orbit the radius does not change. This removes some terms. $\endgroup$ Commented May 16, 2023 at 7:41


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