Derive Lorentz Equation from Relativistic Hamilton-Jacobi Equation

Consider a ralativistic particle of rest mass $$m$$ and electric charge $$e$$ moving in electromagnetic field with four-potential $$A^{\mu}=(\phi ,\mathrm {A} )}$$ 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

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

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

• Suggestion: Replace the word Lorentz equation with the word Lorentz force. Oct 29 '18 at 13:15
• Comments to the post (v6): 1. Eq. (6) seems wrong for dimensional reasons. 2. Double-check indices in eq.(7). Oct 30 '18 at 14:46

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.