Lagrange function, (the kinetic potential), the characteristic function L(qi, q`i, t) of a mechanical system, expressed through the generalised coordinates qi, the generalised velocities q`i and the time t. In the simplest case of a conservative system the Lagrange function is equal to the difference between the kinetic energy T and the potential energy V of the system, expressed through qi and q`i, that is L=T(qi, q`i,t) -Vqi;. Knowing the Lagrange function, one can use the principle of least action to write down the differential equations of motion of a mechanical system.
The concept of the “Lagrange function” extends also to systems with an infinite number of degrees of freedom — the classical physical fields; here the generalised coordinates and momenta are the values of the field function and their time derivatives at each point of space-time. As in classical mechanics, the Lagrange function determines the equations of motion of the field through the principle of least action. An important property of the Lagrange function is the relativistic invariance of its density (the value of the Lagrange function per unit volume of the field), along with its other symmetry properties. To each of the symmetries there corresponds a conservation law for some physical quantity. Thus invariance under gauge symmetry corresponds to the conservation of charge, and so on. (seeConservation laws).
The concept of the “Lagrange function” extends also to systems with an infinite number of degrees of freedom — the classical physical fields; here the generalised coordinates and momenta are the values of the field function and their time derivatives at each point of space-time. As in classical mechanics, the Lagrange function determines the equations of motion of the field through the principle of least action. An important property of the Lagrange function is the relativistic invariance of its density (the value of the Lagrange function per unit volume of the field), along with its other symmetry properties. To each of the symmetries there corresponds a conservation law for some physical quantity. Thus invariance under gauge symmetry corresponds to the conservation of charge, and so on. (seeConservation laws).