# Relativistic Harmonic Oscillator Lagrangian and Four Force

• GL_Black_Hole
In summary, the conversation discusses the Lagrangian for a relativistic harmonic oscillator in an inertial laboratory frame, and the properties of its canonical momentum and four-momentum. It also explores the Hamiltonian and four-force in the lab frame, as well as the oscillation of mass with a position-dependent angular frequency in the proper time frame.
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

## 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.

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!

## 1. What is the Relativistic Harmonic Oscillator Lagrangian?

The Relativistic Harmonic Oscillator Lagrangian is a mathematical expression used in the study of relativistic mechanics. It describes the behavior of a particle undergoing harmonic motion in a relativistic framework, taking into account the effects of special relativity.

## 2. How is the Relativistic Harmonic Oscillator Lagrangian derived?

The Relativistic Harmonic Oscillator Lagrangian is derived by applying the principles of Lagrangian mechanics to a system that follows the laws of special relativity. This involves considering the kinetic and potential energy of the particle in a relativistic context, and then using the Euler-Lagrange equations to find the equations of motion.

## 3. What is the significance of the Relativistic Harmonic Oscillator Lagrangian in physics?

The Relativistic Harmonic Oscillator Lagrangian is significant because it allows us to accurately model and understand the behavior of particles in a relativistic framework. This is important in fields such as particle physics and cosmology, where relativistic effects play a major role.

## 4. What is the Four Force of the Relativistic Harmonic Oscillator Lagrangian?

The Four Force is a vector quantity that represents the combined effects of all four fundamental forces (gravity, electromagnetism, strong nuclear force, and weak nuclear force) on a particle in a relativistic harmonic oscillator system. It is derived from the Relativistic Harmonic Oscillator Lagrangian and can be used to calculate the acceleration of the particle.

## 5. How does the Relativistic Harmonic Oscillator Lagrangian differ from the classical Harmonic Oscillator Lagrangian?

The Relativistic Harmonic Oscillator Lagrangian takes into account the effects of special relativity, such as time dilation and length contraction, which are not present in the classical Harmonic Oscillator Lagrangian. This makes the equations of motion and the resulting behavior of the system significantly different in the two frameworks.

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