Question

The Lagrangian for a \(\pi\)-meson is given by $$ L(\mathbf{x}, t)=\frac{1}{2}\left(\dot{\phi}^{2}-|\nabla \phi|^{2}-\mu^{2} \phi^{2}\right) $$ where \(\mu\) is the meson mass and \(\phi(\mathbf{x}, t)\) is its wavefunction. Assuming Hamilton's principle find the wave equation satisfied by \(\phi\).

Short answer

\(\abla^2 \phi - \frac{\partial^2 \phi}{\partial t^2} + \mu^2 \phi = 0.\)

Step-by-step solution

Identify the Lagrangian

The given Lagrangian for the \( \pi\ \)-meson is \(\frac{1}{2}\big(\frac{\text{d}\big)}{} - |abla \phi|^2 - \mu^2 \phi^2\big)\). Here, \(\phi(\mathbf{x}, t)\) is the wavefunction and \(\mu\) is the meson mass.Labor pain is labor pain.

Use Euler-Lagrange Equation

Hamilton's principle states that the action \(S = \int L \, \text{d}t \) is stationary. The Euler-Lagrange equation for \(\phi\) is given by \(\frac{\partial L}{\partial \phi} - \frac{\text{d}}{\text{d}t} \big(\frac{\partial L}{\partial \dot{\phi}}\big) = 0.\)

Compute Partial Derivatives

Find the partial derivatives: \(\frac{\partial L}{\partial \phi} = -\mu^2 \phi,\)\(\frac{\partial L}{\partial \dot{\phi}} = \dot{\phi},\) \ and \(\frac{\partial L}{\partial (\abla \phi)} = -\abla \phi.\)

Apply the Euler-Lagrange Equation

Substitute the partial derivatives into the Euler-Lagrange equation: \(\abla^2 \phi - \frac{\partial^2 \phi}{\partial t^2} + \mu^2 \phi = 0.\)This simplifies to the Klein-Gordon equation.

Key concepts

Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics introduced by Joseph-Louis Lagrange in the late 18th century. It's a vital part of theoretical physics. Instead of using forces as in Newtonian mechanics, Lagrangian mechanics focuses on energy.

The core idea is the Lagrangian, denoted as \( L \), which is a function that summarizes the dynamics of a system. For a system with generalized coordinates \( q_i \) and their time derivatives \( \dot{q_i} \), the Lagrangian is usually defined as:
\[ L = T - V \] where:

• \( T \) is the kinetic energy
• \( V \) is the potential energy

In more complex cases, the Lagrangian can incorporate additional fields and variables. In the context of a \( \pi \)-meson, the Lagrangian provided is:

\[L(\textbf{x}, t) = \frac{1}{2} \left( \dot{\phi}^{2} - |\abla \phi|^{2} - \mu^{2} \phi^{2} \right)\]
Here:

• \( \phi(\textbf{x}, t) \) is the wavefunction
• \( \mu \) is the meson mass

This setup leads us into applying principles like Hamilton's principle.

Euler-Lagrange equation

The Euler-Lagrange equation is essential for deriving equations of motion in Lagrangian mechanics. It comes from Hamilton's principle which states that the true path taken by a system is the one that minimizes (or makes stationary) the action \( S \). The action is given by the integral of the Lagrangian over time:
\[ S = \int L \left( q_i, \dot{q_i}, t \right) \text{d}t \]
The Euler-Lagrange equation can be expressed as:
\[ \frac{\partial L}{\partial q_i} - \frac{\text{d}}{\text{d}t} \left( \frac{\partial L}{\partial \dot{q_i}} \right) = 0 \]
In the given exercise, they compute partial derivatives for \( \phi \):

• \( \frac{\partial L}{\partial \phi} = -\mu^2 \phi \)
• \( \frac{\partial L}{\partial \dot{\phi}} = \dot{\phi} \)
• \( \frac{\partial L}{\partial (\abla \phi)} = -\abla \phi \)

Substituting these back into the Euler-Lagrange equation results in the wave equation.

Wave equation

Wave equations describe the propagation of waves through a medium. They are a fundamental part of physics, appearing in various contexts, such as acoustics, electromagnetics, and quantum mechanics.

In the context of the \( \pi \)-meson problem, after applying the Euler-Lagrange equation, we get to the Klein-Gordon equation, which is a type of wave equation for relativistic particles:
\[ \abla^{2} \phi - \frac{\partial ^{2} \phi}{\partial t^{2}} + \mu ^{2} \phi = 0 \]
This equation describes how the wavefunction \( \phi \) evolves through space and time. It incorporates the mass term \( \mu \) which is characteristic of the particle it describes. Understanding this wave equation is crucial for dealing with quantum fields and particles in advanced physics.

By studying these equations, we learn how particles behave and propagate, providing essential insights into the nature of matter and energy.