Relativistic Harmonic Oscillator Lagrangian and Four Force

[GL_Black_Hole]

Homework Statement

Consider an inertial laboratory frame S with coordinates ($\lambda$; $x$). The Lagrangian for the
relativistic harmonic oscillator in that frame is given by
$L =-mc\sqrt{\dot x^{\mu} \dot x_{\mu}} -\frac {1}{2} k(\Delta x)^2 \frac{\dot x^{0}}{c}$ where $x^0 =c\lambda$, $x^1 =x$.
a) Find the canonical momentum $\Pi _{\mu} = \frac{\partial L}{\partial \dot x^{\mu}}$ in the lab frame ($t$;$x$). Are any of its components conserved? Is the canonical momentum a four vector?
b) Find the components of the four-momentum $p^{\mu} = m \frac{d\dot x^{\mu}}{d\tau}$
c) Find the Hamiltonian in the lab frame using a Legendre transform and show that it is conserved in the lab frame. Now are either $c \Pi^{0}$ or $c p^{0}$ conserved?
d) Show that the four force $F^{\mu}$ satisifies $F^{\mu} =-\frac{k\Delta x}{c} \epsilon^{\mu \nu} \dot x_{\nu}$
e) Now go to the proper time frame ($\tau$, $x$) and show that the mass oscillates with a position dependent angular frequency $\omega (x)$
Note: the proper time frame is NOT the rest frame the mass. It only has clocks which follow along with the proper time of the particle

Homework Equations

The Attempt at a Solution

a) In the lab frame the Lagrangian is $L= \frac{-mc^2}{\gamma} -\frac{1}{2} k(\Delta x)^2.$ I find that $\dot x_{0} = -c$ so then $\Pi _{0} = -\frac{1}{c}({-mc^2}{\gamma} -\frac{1}{2} k(\Delta x)^2) =\frac{E}{c}$ and $\Pi_{1} = \gamma {m\dot x}$ So the components of $\Pi^{\mu}$ are $\Pi^{0} = \frac{-E}{c}$ and $\Pi^{1} =\gamma {m\dot x}$. The 0 component is conserved but because the canonical momentum is formed by taking a derivative with respect to non-invariant quantity, coordinate time, it cannot be a four vector.
b) Here I'm not sure if I'm missing something: $p^{0} = m \frac{d\dot x^{0}}{d\tau} = mc\gamma$ because $\frac{dt}{d\tau} = \gamma$ So similarly, $p^{1} = mv\gamma$ where $v =\frac{dx}{dt}$.
c) I use the canonical momentum here and form $H =\Pi^{ \mu}\dot x_{\mu} -L$ and so $H = -\frac{1}{c}(\gamma mc^2 + \frac{1}{2} k(\Delta x)^2)(-c) + (\gamma mv)(v) -L$ but evaluating this gives me: $(\gamma +\frac{1}{\gamma})mc^2 + \gamma mv^2 + k(\Delta x)^2$ which does not equal E as it should.

 

[Nate810]

d) Here I'm not sure what to do and any help would be very much appreciated! e) I know that in the proper time frame $x^{\mu} = x^{\mu} (\tau)$ so $\frac{dx^{\mu}}{d\tau} = \dot x^{\mu}$ and the Lagrangian is $L= -mc^2 - \frac{1}{2}k(\Delta x)^2 \dot x^{0}$. I find that the canonical momentum $\Pi_{\mu} = \frac{\partial L}{\partial \dot x^{\mu}} = -mc\gamma \delta_{\mu}^{0}$ and so $p^{\mu} = m \frac{d\dot x^{\mu}}{d\tau} = m \frac{d(-mc\gamma \delta_{\mu}^{0})}{d\tau} = -mk\gamma \Delta x \epsilon^{\mu \nu}\dot x_{\nu}$ which is not quite the same as the required result. Thanks in advance for reading this and any help is really appreciated!