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Mathematics for Physics

108 Derivation of Lagrangian

A derivation that is used to obtain \(L=T-V\) as the slow-motion limit of a relativistic free particle with a potential, where the Lagrangian is the function whose stationary action gives the equations of motion.

1. A free particle with a potential has a Lorentz-invariant Lagrangian built from rest energy, speed, and potential energy. Rest energy is the energy \(mc^{2}\) of a particle at rest. This principle is used to start from a relativistic expression that reduces to Newtonian mechanics at low speed.

The relativistic Lagrangian is

\[ L_{\mathrm{rel}} = -m c^{2}\sqrt{1 - \dfrac{1}{c^{2}}\left(\dfrac{dx}{dt}\right)^{2}} - V(x) \]

where

  • \(L_{\mathrm{rel}}\) is the relativistic Lagrangian.
  • \(m\) is the mass.
  • \(c\) is the speed of light.
  • \(x\) is the position.
  • \(t\) is time.
  • \(V(x)\) is the potential energy.

2. For a slow particle the square root expands to first order in the square of the speed over \(c\). A slow particle is a particle whose speed is much smaller than the speed of light. This principle is used to extract the Newtonian kinetic term.

The slow-motion expansion is

\[ \sqrt{1 - \dfrac{1}{c^{2}}\left(\dfrac{dx}{dt}\right)^{2}} \approx 1 - \dfrac{1}{2c^{2}}\left(\dfrac{dx}{dt}\right)^{2} \]

where

  • \(c\) is the speed of light.
  • \(\dfrac{dx}{dt}\) is the velocity.

3. Substituting the expansion and dropping the constant rest-energy term produces the non-relativistic Lagrangian. This principle is used to obtain \(L=T-V\) for the Schrödinger Hamiltonian and for the path integral.

The non-relativistic Lagrangian is

\[ L = T - V = \dfrac{1}{2}m\left(\dfrac{dx}{dt}\right)^{2} - V(x) \]

where

  • \(L\) is the non-relativistic Lagrangian.
  • \(T\) is the kinetic energy.
  • \(V\) is the potential energy.
  • \(m\) is the mass.

4. In quantum mechanics the same \(L\) supplies the phase \(e^{iS/\hbar}\) of each path. This principle is used to connect the classical Lagrangian to quantum amplitudes.

Note: These principles are the relativistic Lagrangian, the slow-motion expansion, \(L=T-V\), and the path-integral phase.

108.1 References

  1. Schwichtenberg, J. Physics from Symmetry. Springer, 2018. — from \(L_{\mathrm{rel}}\) to \(L=T-V\).
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(L=T-V\) in the path-integral formulation.