Difficulty in understanding Newton first law in case of magnetic field

Question

I am Math Undergrad student, reading this article. In that Author mentioned following

Consider the motion a charged particle of unit mass and unit charge in this magnetic field, which is described by Newton's law of motion $$ \nabla_{\dot{\gamma}} \dot{\gamma}=Y(\dot{\gamma}) $$ where $\nabla$ is the Levy-Civita connection of $g$ and $Y: T M \rightarrow T M$ is the Lorentz force associated with $\Omega$, i.e., the bundle map uniquely determined by $$ \Omega_{x}(\xi, \eta)=\left\langle Y_{x}(\xi), \eta\right\rangle_{g} $$ for all $x \in M$ and $\xi, \eta \in T_{x} M .$

I do not understand how Newton first law implies $$ \nabla_{\dot{\gamma}} \dot{\gamma}=Y(\dot{\gamma}) $$

Also I do not understand Lorentz force $Y$ Associated with $\Omega$.

I do not any background on Physics. So if anyone explained to me above how it follows or some reference that would be really nice.

Thank you so much.

Answer 1

The expression $$\nabla_{\dot \gamma}\dot \gamma = Y(\dot \gamma) \tag{$\star$}\label{$\star$}$$ is Newton's second law. Newton's first law - which states that objects which are not under the influence of external forces follow geodesics - could be understood as a limiting case of $\eqref{$\star$}$ in which $Y=0$.

As far as the relationship between $\Omega$ and $Y$, we have that

$$\Omega(\xi,\eta) = \langle Y(\xi),\eta\rangle_g$$ $$\implies \Omega_{ij} \xi^i \eta^j = g_{\alpha j} Y^\alpha_{\ \ i}\xi^i \eta^j \iff Y^\alpha_{\ \ \ i} = g^{\alpha j} \Omega_{ij}$$ So the right hand-side says "plug the vector $\dot \gamma$ into the first slot of $\Omega$ to get a covector, then raise the index with $g$ to get a vector."

In three dimensions, $\Omega$ is related to the standard magnetic field $B$ (a pseudo-vector field) via $\Omega_{ij} = -\epsilon_{ijk} B^k$, and so we could also write

$$Y^\alpha_{\ \ \ i} = -\epsilon_{ijk} g^{\alpha j} B^k\implies [Y(\dot \gamma)]^\alpha = -g^{\alpha j}\epsilon_{jki} B^k \dot\gamma^i \equiv (\dot \gamma \times \vec B)^\alpha$$ where $\times$ denotes the ordinary vector cross-product; this therefore reproduces the familiar (to physicists) magnetic force law $$\vec F = q(\vec v \times \vec B)$$