213 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 from the relativistic Lagrangian.
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 produces a constant rest-energy term plus \(\dfrac{1}{2}m\left(\dfrac{dx}{dt}\right)^{2}\) minus the potential. Kinetic energy is the energy of motion. This principle is used to identify the Newtonian kinetic energy.
The expanded Lagrangian is
\[ L_{\mathrm{rel}} \approx -m c^{2} + \dfrac{1}{2}m\left(\dfrac{dx}{dt}\right)^{2} - V(x) \]
where
- \(L_{\mathrm{rel}}\) is the relativistic Lagrangian.
- \(m\) is the mass.
- \(c\) is the speed of light.
- \(\dfrac{dx}{dt}\) is the velocity.
- \(V(x)\) is the potential energy.
4. An additive constant in the Lagrangian does not change the Euler-Lagrange equation, so the rest-energy term \(-mc^{2}\) may be dropped. This principle is used to pass to the standard non-relativistic Lagrangian.
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.
- \(\dfrac{dx}{dt}\) is the velocity.
5. The Euler-Lagrange equation applied to \(L=T-V\) recovers Newton’s second law. This principle is used to confirm that the derived Lagrangian describes the same Newtonian motion.
The Euler-Lagrange recovery of Newton’s law is
\[ m\dfrac{d^{2}x}{dt^{2}} = -\dfrac{dV}{dx} \]
where
- \(m\) is the mass.
- \(x\) is the position.
- \(t\) is time.
- \(V\) is the potential energy.
Note: These principles are the relativistic free-particle Lagrangian with a potential, the slow-motion expansion, the identification of kinetic energy, the irrelevance of an additive constant in the Lagrangian, and the recovery of Newton’s second law from the Euler-Lagrange equation.