Relativistic Lagrangian mechanics

Source: ChatGPT.
Relativistic Lagrangian mechanics is an extension of classical Lagrangian mechanics to systems where the effects of special relativity are significant. It incorporates the principles of relativity, ensuring that the laws of physics are invariant under Lorentz transformations.

In the relativistic regime, the Lagrangian is constructed using Lorentz-invariant quantities, particularly focusing on the spacetime interval, which remains invariant across different inertial reference frames.

Case of a Free Particle in Minkowski space

For a free particle of mass $m$ moving in flat spacetime:

1. Spacetime Interval: The proper time $\tau$ is used to describe the motion of the particle. The proper time interval $d\tau$ is related to the spacetime coordinates by:

   $$d\tau =\sqrt{1-\frac{{\mathbf{v}}^2}{c^2}}dt=\frac{1}{c}\sqrt{-d{s}^2},$$

   where $d{s}^2=-c^{2}d{t}^2+d{x}^2+d{y}^2+d{z}^2$ is the Minkowski spacetime interval.

2. Action and Lagrangian: The action $S$ is proportional to the proper time:

   $$S=-m{c}^2\int d\tau .$$

   Expressing $d\tau$ in terms of $dt$, the action becomes:

   $$S=-m{c}^2\int \sqrt{1-\frac{{\mathbf{v}}^2}{c^2}}dt.$$

   The relativistic Lagrangian $L$ for the free particle is:

   $$L=-m{c}^2\sqrt{1-\frac{{\mathbf{v}}^2}{c^2}}.$$

In the non-relativistic limit ($v\ll c$), the Lagrangian reduces to:

where the $-m{c}^2$ term corresponds to the rest energy, and the $\frac{1}{2}m{\mathbf{v}}^2$ term is the familiar kinetic energy.

This formulation sets the foundation for studying relativistic systems, where velocities approach the speed of light or spacetime curvature effects (in general relativity) are incorporated.
From here we can go to relativistic Hamiltonian mechanics.