The Lagrangian of a system is defined as
where T is the total kinetic energy and V is the total potential energy. Given a Lagrangian L, consider
where q is a generalized coordinate and is its time derivative. Then
so
This means that the two Lagrangians represent the same equation of motion with
Hamiltonian, Lagrangian Coordinates, Lagrangian Mechanics, Lagrangian Turbulence
References
Goldstein, H. Ch. 11 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.
Jackson, J. D. "Lowest-Order Relativistic Corrections to the Lagrangian for Interacting Charged Particles, the Darwin Lagrangian." §12.7 in Classical Electrodynamics, 3rd ed. New York: Wiley, pp. 593-595, 1998.
Landau, L. D. and Lifschitz, E. M. "The Lagrangian to Terms of Second Order." §65 in The Classical Theory of Fields, 4th ed. Oxford, England: Pergamon Press, pp. 165-169, 1987.
Wentzel, G. Ch. 1 in Quantum Theory of Fields. New York: Interscience Publishers, 1949.
