homework and exercises - Special Relativity - Test Particle inside the Sun's Gravitational Field - Differential Equation

Question

[Question Context: Consider the motion of a test particle of (constant) mass $m$ inside the gravitational field produced by the Sun in the context of special relativity.

In addition, consider the equations of motion for the test particle, which can be written as

$$\frac{d(m\gamma c)}{dt} = \frac{\vec{v}}{c} \cdot \vec{F}$$

OR

$$\frac{d(m\gamma \vec{v})}{dt} = \vec{F},$$

where $\vec{v}$ is the speed of the test particle, $c$ is the (constant) speed of light, and by definition,

$$\gamma \equiv \frac{1}{\sqrt{1- \frac{\vec{v}^2}{c^2}}} .$$

In addition, the gravitational force is given by

$$\vec{F} \equiv -\frac{GMm}{r^2} \hat{e}_r$$

where $\hat{e}_r$ is the unit vector in the direction between the Sun (of mass $M$) and the test particle (of mass $m$).]

The Question Itself

Solve the previously found differential equation

$$\frac{d^2u}{d\theta^2} + u \bigg( 1- \frac{G^2 M^2}{\ell^2 c^2} \bigg) - \frac{GMd}{\ell^2} = 0$$ for the trajectory, i.e. find the solution for $u(θ)$ (for all $θ$). What kind of trajectories do you find?

Source: [NOT APPLICABLE]

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Personal Comment

Perhaps it's just me, however, I can't seem to solve this differential equation in a clean manner. For some reason, I always get a ton of constants and I feel like I am doing something wrong. With that in mind, any assistance, hints, or comments to help me toward the right answer would be much appreciated. Thank you for reading!

Answer

Re-write for legibility:

$$u''(\theta)+\alpha u(\theta)-\beta=0$$ Make a substitution: $$y=\alpha u(\theta)-\beta$$ So: $$y'=\alpha u'$$ And: $$y''=\alpha u''$$ $$\Rightarrow u''=\frac{y''}{\alpha}$$ $$\Rightarrow \frac{y''}{\alpha}+y=0$$ $$y''(\theta)+\alpha y(\theta)=0$$

Which is the classic ODE of the SHM. Solve and back-substitute. Don't neglect the BCs!