What is a Lagrangian?
Short Answer
A Lagrangian is a function that encodes the dynamics of a physical system. It is the difference between kinetic and potential energy in classical mechanics, and generalizes to describe field theories in QFT.
Classical Mechanics
For a particle with position q(t), the Lagrangian is:
L(q, \dot{q}, t) = T - V = \frac{1}{2}m\dot{q}^2 - V(q)
where:
- T = kinetic energy
- V = potential energy
The Action Principle
The action is the integral of the Lagrangian over time:
S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt
Physical trajectories are those that make the action stationary (typically a minimum):
\delta S = 0
This leads to the Euler-Lagrange equations:
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0
Field Theory
For fields \phi(x), the Lagrangian becomes a Lagrangian density \mathcal{L}:
S[\phi] = \int \mathcal{L}(\phi, \partial_\mu \phi) \, d^4x
Example: Free Scalar Field
The Klein-Gordon Lagrangian for a free scalar field:
\mathcal{L} = \frac{1}{2}\partial_\mu \phi \, \partial^\mu \phi - \frac{1}{2}m^2 \phi^2
Example: Electromagnetism
The Maxwell Lagrangian:
\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
where F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu is the field strength tensor.
Why Lagrangians Matter
- Symmetries: Noether’s theorem connects symmetries of L to conserved quantities
- Quantization: Path integrals use e^{iS/\hbar} as the quantum amplitude
- Gauge theories: Lagrangians naturally encode gauge symmetry
- Renormalization: The Lagrangian determines which terms are allowed
References
- L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Pergamon, 1976), Chapters 1–2
- H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison-Wesley, 2002)
- V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, 1989)
- M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995), Chapter 2
- D. Tong, Lectures on Classical Dynamics — Free online notes