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Mathematics for Physics

123 Lagrangian

A function that is used to encode the difference between kinetic energy and potential energy for the equations of motion and for the phase of a quantum path.

\(L=T-V\). The Lagrangian is kinetic energy minus potential energy. This principle is used to write \(L\) for a particle in a potential.

The Lagrangian is

\[ L = T - V \]

where

  • \(L\) is the Lagrangian.
  • \(T\) is the kinetic energy.
  • \(V\) is the potential energy.

The generalized-coordinate Lagrangian. In generalized coordinates, \(L\) is a function of the coordinates, the velocities, and possibly time. This principle is used to write the Euler-Lagrange equations.

The Lagrangian in generalized coordinates is

\[ L = L\Bigl(t,\, q_{1},\ldots,q_{m},\, \dfrac{dq_{1}}{dt},\ldots,\dfrac{dq_{m}}{dt}\Bigr) \]

where

  • \(q_{1},\ldots,q_{m}\) are generalized coordinates.
  • \(\dfrac{dq_{i}}{dt}\) are generalized velocities.
  • \(t\) is time.

Stationary action. The action is the time integral of \(L\), and the classical path makes the action stationary. This principle is used to derive the equations of motion.

The action is

\[ S = \displaystyle\int_{t_{i}}^{t_{f}} L\,dt \]

where

  • \(S\) is the action.
  • \(L\) is the Lagrangian.
  • \(t\) is time.

The path-integral phase. In the path-integral formulation, each path contributes a phase \(e^{iS/\hbar}\). This principle is used to compute quantum amplitudes from the same \(L\) that governs classical motion.

Note: Also denoted \(L\).

123.1 References

  1. Park, D. Introduction to the Quantum Theory. Dover, 2005. — \(L=T-V\) with \(T\) kinetic and \(V\) potential energy of the whole system.
  2. Simmons, G. F. Differential Equations with Applications and Historical Notes. Chapman and Hall/CRC, 2017. — \(L=T-V\) as a function of \(t\), \(q_{j}\), and \(\mathrm{d}q_{j}/\mathrm{d}t\).
  3. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(L(q_{a},\mathrm{d}q_{a}/\mathrm{d}t;t)=T-V\).
  4. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(L=T-V\) in the path-integral formulation.