Derive Lorentz Equation from Relativistic Hamilton-Jacobi Equation

Question

Consider a ralativistic particle of rest mass $m$ and electric charge $e $ moving in electromagnetic field with four-potential ${\displaystyle A^{\mu}=(\phi ,\mathrm {A} )} $ in vacuum, then the Hamilton–Jacobi equation has the form

$$g^{\mu \nu}\left ( \frac{\partial S}{\partial x^{\mu}} + \frac {e}{c}A_{\mu} \right ) \left ( \frac{\partial S}{\partial x^{\nu}} + \frac {e}{c}A_{\nu} \right ) = m^2 c^2\tag{1}$$

or more compact expressed as Minkowski product

$$ \left( \frac{\partial S}{\partial x^{\mu}} + \frac {e}{c}A_{\mu} \right ) \left ( \frac{\partial S}{\partial x_{\mu}} + \frac {e}{c}A^{\mu} \right ) = m^2 c^2 \tag{2}$$

here we denote $g^{\mu \nu}$ the metric tensor with signature $(+ - - -)$ and $S$ is the action function from Hamilton-Jacobi-theory.

Especially $S$ satisfy the equation

$$p_{\mu}= \nabla_{\mu}S := \frac{\partial S}{\partial x^{\mu}}\tag{3}$$

where $p_{\mu}$ is the four momentum and $\nabla_{\mu}$ the four gradient.

Now I have following two questions:

1. Does anybody have a reference for a rigorous derivation for $$ \left( \frac{\partial S}{\partial x^{\mu}} + \frac {e}{c}A_{\mu} \right ) \left ( \frac{\partial S}{\partial x_{\mu}} + \frac {e}{c}A^{\mu} \right ) = m^2 c^2 .\tag{4}$$

2. It is known that applying method of characteristics to the PDE

$$F(S,\frac{\partial S}{\partial x^{\mu}} ,x^{\mu}):= \left( \frac{\partial S}{\partial x^{\mu}} + \frac {e}{c}A_{\mu} \right ) \left ( \frac{\partial S}{\partial x_{\mu}} + \frac {e}{c}A^{\mu} \right ) - m^2 c^2 =0\tag{5}$$

one can derive the relative Lorentz equation

$${\displaystyle {\frac {\mathrm {d} p^{\mu }}{\mathrm {d} \tau }}=eF^{\mu \nu }p_{\nu }}\tag{6}$$

with electromagnetic tensor $$F^{\mu \nu }:= \frac{\partial A_{\mu}}{\partial x^{\nu}}- \frac{\partial A_{\nu}}{\partial x^{\mu}}\tag{7}$$ and four momentum $p_{\mu}$.

Here I'm also looking for an explicit derivation of LE from the HJE using characteristics.

Indeed, the method of characteristics transform a PDE into a system of ODE with respect parametrizing variable $\tau$:

$$\frac{\partial p_{\mu}}{\partial \tau}= -\frac{\partial F}{\partial x^{\mu}} -\frac{\partial F}{\partial S} p_{\mu}\tag{8}$$

$$\frac{\partial x_{\mu}}{\partial \tau}= \frac{\partial F}{\partial p^{\mu}}. \tag{9}$$

Remark: HJ theory says $$p_{\mu}= \frac{\partial S}{\partial x^{\mu}}.\tag{10}$$

The problem is to derive from here the equation for Lorentz force

Answer 1

This answer does not address OP's specific question about the method of characteristics, but sketches a systematic derivation (of the various equations involved) starting from a Lagrangian formulation.

1. A Lagrangian for a relativistic point particle of mass $m$ and charge $q$ in a EM background $A_{\mu}$ and gravitational background $g_{\mu\nu}$ is$^1$ $$ L~:=~L_0 - U,\qquad L_0~:=~\pm \frac{\dot{x}^2}{2e}-\frac{e m^2}{2}, \qquad \dot{x}^2~:=~g_{\mu\nu}(x)~ \dot{x}^{\mu}\dot{x}^{\nu}, \qquad \dot {x}^{\mu} ~:=~\frac{dx^{\mu}}{d\tau},\tag{A}$$ with Minkowski sign convention $(\mp,\pm,\pm,\pm)$ and speed-of-light $c=1$. Here $\tau$ is the world-line (WL) parameter (which is not necessarily proper time) and $e>0$ is an einbein field. The velocity-dependent Lorentz potential is $$ U~:=~ \mp q{\dot x}^{\mu} A_{\mu}, \tag{B} $$ with corresponding generalized Lorentz 4-force$^2$ $$ F_{\mu}~:=~\frac{d}{d\tau} \frac{\partial U}{\partial \dot{x}^{\mu}} - \frac{\partial U}{\partial x^{\mu}}~=~\pm qF_{\mu\nu}\dot {x}^{\nu}, \qquad F_{\mu\nu}~:=~\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}. \tag{C}$$

2. The canonical/conjugate 4-momentum$^2$ becomes $$ p_{\mu}~:=~\frac{\partial L}{\partial \dot{x}^{\mu}}~=~\pm\frac{g_{\mu\nu}\dot{x}^{\nu}}{e}\pm q A_{\mu}.\tag{D}$$

3. After a Legendre transformation the Hamiltonian reads $$ H~=~\frac{e}{2}\left(m^2\pm ( p\mp qA)^2 \right). \tag{E}$$ Note that the $e$ field is a Lagrange multiplier for the mass-shell constraint.

4. The Hamilton-Jacobi (HJ) equation becomes essentially the mass-shell constraint $$ 0~=~E~=~\frac{e}{2}\left(m^2\pm ( \frac{\partial W}{\partial x}-qA)^2 \right), \tag{F}$$ where $W$ is Hamilton's characteristic function, and$^3$ $$ p_{\mu}~=~\pm \frac{\partial W}{\partial x^{\mu}}. \tag{G}$$ The fact that the energy $E$ is zero can be viewed as a consequence of WL reparametrization invariance $\tau\to \tau^{\prime}=f(\tau)$.

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$^1$ To achieve the standard square root Lagrangian, simply integrate out the $e$ field, cf. e.g. this Phys.SE post.

$^2$ The usual notions of 4-momentum & 4-force correspond to the gauge where the world-line (WL) parameter $\tau$ is proper time.

$^3$ For sign conventions, see also this Phys.SE post.