Peeter Joot's (OLD) Blog.

Math, physics, perl, and programming obscurity.

Non-covariant Lagrangian and Hamiltonian for Lorentz force.

Posted by peeterjoot on November 28, 2009

[Click here for a PDF containing this post with nicer formatting]

In [1], the Lagrangian for a charged particle is given as (12.9) as

\begin{aligned}\mathcal{L} = -m c^2 \sqrt{1 - \mathbf{u}^2/c^2} + \frac{e}{c} \mathbf{u} \cdot \mathbf{A} - e \Phi.\end{aligned} \quad\quad\quad(1)

Let’s work in detail from this to the Lorentz force law and the Hamiltonian and from the Hamiltonian again to the Lorentz force law using the Hamiltonian equations. We should get the same results in each case, and have enough details in doing so to render the text a bit more comprehensible.

Canonical momenta

We need the conjugate momenta for both the Euler-Lagrange evaluation and the Hamiltonian, so lets get that first. The components of this are

\begin{aligned}\frac{\partial {\mathcal{L}}}{\partial {\dot{x}_i}} &= - \frac{1}{{2}} m c^2 \gamma (-2/c^2) \dot{x}_i + \frac{e}{c} A_i \\ &= m \gamma \dot{x}_i + \frac{e}{c} A_i.\end{aligned}

In vector form the canonical momenta are then

\begin{aligned}\mathbf{P} &= \gamma m \mathbf{u} + \frac{e}{c} \mathbf{A}.\end{aligned} \quad\quad\quad(2)

Euler-Lagrange evaluation.

Completing the Euler-Lagrange equation evaluation is the calculation of

\begin{aligned}\frac{d\mathbf{P}}{dt} = \boldsymbol{\nabla} \mathcal{L}.\end{aligned} \quad\quad\quad(3)

On the left hand side we have

\begin{aligned}\frac{d\mathbf{P}}{dt} = \frac{d(\gamma m \mathbf{u})}{dt} + \frac{e}{c} \frac{d\mathbf{A} }{dt},\end{aligned} \quad\quad\quad(4)

and on the right, with implied summation over repeated indexes, we have

\begin{aligned}\boldsymbol{\nabla} \mathcal{L} = \frac{e}{c} \mathbf{e}_k (\mathbf{u} \cdot \partial_k \mathbf{A}) - e \boldsymbol{\nabla} \Phi.\end{aligned} \quad\quad\quad(5)

Putting things together we have

\begin{aligned}\frac{d(\gamma m \mathbf{u})}{dt} &= -e \left(\boldsymbol{\nabla} \Phi + \frac{1}{{c}} \frac{\partial {\mathbf{A}}}{\partial {t}}+ \frac{1}{c} \left(\frac{\partial {\mathbf{A}}}{\partial {x_a}} \frac{\partial {x_a}}{\partial {t}} - \mathbf{e}_k (\mathbf{u} \cdot \partial_k \mathbf{A}) \right)\right) \\ &= -e \left(\boldsymbol{\nabla} \Phi + \frac{1}{{c}} \frac{\partial {\mathbf{A}}}{\partial {t}}+ \frac{1}{c} \mathbf{e}_b u_a\left(\frac{\partial {A_b}}{\partial {x_a}} -\frac{\partial {A_a}}{\partial {x_b}}\right)\right).\end{aligned}

With

\begin{aligned}\mathbf{E} = -\boldsymbol{\nabla} \Phi - \frac{1}{{c}} \frac{\partial {\mathbf{A}}}{\partial {t}},\end{aligned} \quad\quad\quad(6)

the first two terms are recognizable as the electric field. To put some structure in the remainder start by writing

\begin{aligned}\frac{\partial {A_b}}{\partial {x_a}} - \frac{\partial {A_a}}{\partial {x_b}} = \epsilon^{fab} {(\boldsymbol{\nabla} \times \mathbf{A})}_f.\end{aligned} \quad\quad\quad(7)

The remaining term, with \mathbf{B} = \boldsymbol{\nabla} \times \mathbf{A} is now

\begin{aligned}- \frac{e}{c} \mathbf{e}_b u_a \epsilon^{gab} B_g&=\frac{e}{c} \mathbf{e}_a u_b \epsilon^{abg} B_g \\ &= \frac{e}{c} \mathbf{u} \times \mathbf{B}.\end{aligned}

We are left with the momentum portion of the Lorentz force law as expected

\begin{aligned}\frac{d(\gamma m \mathbf{u})}{dt} = e \left( \mathbf{E} + \frac{1}{c} \mathbf{u} \times \mathbf{B} \right).\end{aligned} \quad\quad\quad(8)

Observe that with a small velocity Taylor expansion of the Lagrangian we obtain the approximation

\begin{aligned}-m c^2 \sqrt{ 1 -\mathbf{u}^2/c^2} \approx - m c^2 \left( 1 - \frac{1}{{2}} \mathbf{u}^2/c^2 \right) = \frac{1}{{2}} m \mathbf{u}^2\end{aligned} \quad\quad\quad(9)

If that is our starting place, we can only obtain the non-relativistic approximation of the momentum change by evaluating the Euler-Lagrange equations

\begin{aligned}\frac{d (m \mathbf{u})}{dt} = e \left( \mathbf{E} + \frac{1}{c} \mathbf{u} \times \mathbf{B} \right).\end{aligned} \quad\quad\quad(10)

That was an exercise previously attempting working the Tong Lagrangian problem set [2].

Hamiltonian.

Having confirmed the by old fashioned Euler-Lagrange equation evaluation that our Lagrangian provides the desired equations of motion, let’s now try it using the Hamiltonian approach. First we need the Hamiltonian, which is nothing more than

\begin{aligned}H = \mathbf{P} \cdot \mathbf{u} - \mathcal{L}\end{aligned} \quad\quad\quad(11)

However, in the Lagrangian and the dot product we have velocity terms that we must eliminate in favor of the canonical momenta. The Hamiltonian remains valid in either form, but to apply the Hamiltonian equations we need H = H(\mathbf{P}, \mathbf{x}), and not H = H(\mathbf{u}, \mathbf{P}, \mathbf{x}).

\begin{aligned}H = \mathbf{P} \cdot \mathbf{u} + m c^2 \sqrt{1 - \mathbf{u}^2/c^2} - \frac{e}{c} \mathbf{u} \cdot \mathbf{A} + e \Phi.\end{aligned} \quad\quad\quad(12)

Or

\begin{aligned}H = \mathbf{u} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A}\right) + m c^2 \sqrt{1 - \mathbf{u}^2/c^2} + e \Phi.\end{aligned} \quad\quad\quad(13)

We can rearrange 2 for \mathbf{u}

\begin{aligned}\mathbf{u} = \frac{1}{{m \gamma}} \left( \mathbf{P} - \frac{e}{c} \mathbf{A} \right),\end{aligned} \quad\quad\quad(14)

but \gamma also has a \mathbf{u} dependence, so this is not complete. Squaring gets us closer

\begin{aligned}\mathbf{u}^2 = \frac{1 - \mathbf{u}^2/c^2}{m^2} {\left( \mathbf{P} - \frac{e}{c} \mathbf{A} \right)}^2,\end{aligned} \quad\quad\quad(15)

and a bit of final rearrangement yields

\begin{aligned}\mathbf{u}^2 = \frac{(c \mathbf{P} - e \mathbf{A})^2}{m^2 c^2 + {\left( \mathbf{P} - \frac{e}{c} \mathbf{A} \right)}^2}.\end{aligned} \quad\quad\quad(16)

Writing \mathbf{p} = \mathbf{P} - e \mathbf{A}/c, we can rearrange and find

\begin{aligned}\sqrt{1 - \mathbf{u}^2/c^2} = \frac{m c }{\sqrt{m^2 c^2 +\mathbf{p}^2}}\end{aligned} \quad\quad\quad(17)

Also, taking roots of 16 we must have the directions of \mathbf{u} and \left( \mathbf{P} - \frac{e}{c} \mathbf{A} \right) differ only by a rotation. From 14 we also know that these are colinear, and therefore have

\begin{aligned}\mathbf{u} = \frac{c \mathbf{P} - e \mathbf{A}}{\sqrt{m^2 c^2 + {\left( \mathbf{P} - \frac{e}{c} \mathbf{A} \right)}^2}}.\end{aligned} \quad\quad\quad(18)

This and 17 can now be substituted into 13, for

\begin{aligned}H = \frac{c}{m^2 c^2 + \mathbf{p}^2} \left({\left(\mathbf{P} - \frac{e}{c} \mathbf{A}\right)}^2 + m^2 c^2 \right)+ e \Phi.\end{aligned} \quad\quad\quad(19)

Dividing out the common factors we finally have the Hamiltonian in a tidy form

\begin{aligned}H = \sqrt{ (c \mathbf{P} - e \mathbf{A})^2 + m^2 c^4 } + e\Phi.\end{aligned} \quad\quad\quad(20)

Hamiltonian equation evaluation.

Let’s now go through the exercise of evaluating the Hamiltonian equations. We want the starting point to be just the energy expression 20, and the use of the Hamiltonian equations and none of what led up to that. If we were given only this Hamiltonian and the Hamiltonian principle

\begin{aligned}\frac{\partial {H}}{\partial {P_k}} &= u_k \\  \frac{\partial {H}}{\partial {x_k}} &= -\dot{P}_k, \end{aligned} \quad\quad\quad(21)

how far can we go?

For the particle velocity we have no \Phi dependence and get

\begin{aligned}u_k &= \frac{c (c P_k -e A_k)}{\sqrt{ (c \mathbf{P} - e \mathbf{A})^2 + m^2 c^4 }}\end{aligned} \quad\quad\quad(23)

This is 18 in coordinate form, one of our stepping stones on the way to the Hamiltonian, and we recover it quickly with our first set of derivatives. We have the gradient part \dot{\mathbf{P}} = -\boldsymbol{\nabla} H of the Hamiltonian left to evaluate

\begin{aligned}\frac{d\mathbf{P}}{dt} = \frac{e (c P_k -e A_k) \boldsymbol{\nabla} A_k }{\sqrt{ (c \mathbf{P} - e \mathbf{A})^2 + m^2 c^4 }} - e \boldsymbol{\nabla} \Phi.\end{aligned} \quad\quad\quad(24)

Or

\begin{aligned}\frac{d\mathbf{P}}{dt} = e \left( \frac{u_k}{c} \boldsymbol{\nabla} A_k - \boldsymbol{\nabla} \Phi \right)\end{aligned} \quad\quad\quad(25)

This looks nothing like the Lorentz force law. Knowing that \mathbf{P} - e\mathbf{A}/c is of significance (because we know where we started which is kind of a cheat), we can subtract derivatives of this from both sides, and use the convective derivative operator d/dt = {\partial {}}/{\partial {t}} + \mathbf{u} \cdot \boldsymbol{\nabla} (ie. chain rule) yielding

\begin{aligned}\frac{d}{dt}(\mathbf{P} - e\mathbf{A}/c) = e \left( -\frac{1}{{c}}\frac{\partial {\mathbf{A}}}{\partial {t}} - \frac{1}{{c}} (\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{A} + \frac{u_k}{c} \boldsymbol{\nabla} A_k - \boldsymbol{\nabla} \Phi \right).\end{aligned} \quad\quad\quad(26)

The first and last terms sum to the electric field, and we seen evaluating the Euler-Lagrange equations that the remainder is u_k \boldsymbol{\nabla} A_k - (\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{A} = \mathbf{u} \times (\boldsymbol{\nabla} \times \mathbf{A}). We have therefore gotten close to the familiar Lorentz force law, and have

\begin{aligned}\frac{d}{dt}(\mathbf{P} - e\mathbf{A}/c) = e \left( \mathbf{E} + \frac{\mathbf{u}}{c} \times \mathbf{B} \right).\end{aligned} \quad\quad\quad(27)

The only untidy detail left is that \mathbf{P} - e \mathbf{A}/c doesn’t look much like \gamma m \mathbf{u}, what we recognize as the relativistically corrected momentum. We ought to have that implied somewhere and 23 looks like the place. That turns out to be the case, and some rearrangement gives us this directly

\begin{aligned}\mathbf{P} - \frac{e}{c}\mathbf{A} = \frac{m \mathbf{u}}{\sqrt{1 - \mathbf{u}^2/c^2}}\end{aligned} \quad\quad\quad(28)

This completes the exercise, and we’ve now obtained the momentum part of the Lorentz force law. This is still unsatisfactory from a relativistic context since we do not have momentum and energy on equal footing. We likely need to utilize a covariant Lagrangian and Hamiltonian formulation to fix up that deficiency.

References

[1] JD Jackson. Classical Electrodynamics Wiley. 2nd edition, 1975.

[2] Dr. David Tong. Classical Mechanics Lagrangian Problem Set 1. [online]. http://www.damtp.cam.ac.uk/user/tong/dynamics/mf1.pdf.

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