- #1

GL_Black_Hole

- 21

- 0

## 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.