Lagrangian

The Lagrangian of a physical system is an important function allowing the equations of motion of dynamic variables in the system to be described. Initially it was introduced in analytical mechanics by Joseph Louis Lagrange using the equations that bear his name. A simple example will be sufficient to understand the meaning of these equations.

Consider a set of N particles in a gas which will then have 3N position coordinates $q_{i}$ and 3N velocity coordinates $\dot{q_{i}}$ where the dot denotes the total derivative of the preceding coordinate with respect to time, i.e. $\dot{q_{i}}=\frac{dq_{i}}{dt}$ .

It is known that in the Lagrange formulation of the equations of the mechanics of a particle system, the differential equations of motion of these particles will be given by

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q_i}})- \frac{\partial L}{\partial q_i}=0$

where the so-called Lagrange function $L$ for these particles in a potential V is written

$L (q_{i}, \dot{q_{i}})=\sum_{i=1}^{3n} \frac{m_{i} {\dot{q_{i}}}^2}{2} - V(q_{i})$

The masses $m_i$ obviously being the same for a particle and its three position coordinates. The first term on the right in the equation is a sum of kinetic energies.

Joseph Louis Lagrange (25 January 1736, Turin - 10 April 1813, Paris)

The Lagrangian formulation is invariant to a change in the coordinates of the system $q_{i}$ which is what makes it so powerful, as it allows us to reduce large classes of differential equations expressed in Cartesian, spherical etc. coordinates to a few fundamental cases that can be solved.

Moreover, it can be used to prove that the theorems of conservation of energy, momentum etc. are automatically valid for these large classes of differential equations since, by proving conservation laws in a given coordinate system, it is certain that they will be valid in other coordinate systems or in other mechanical systems that can be cast into Lagrangian form equations.

The validity of the laws of conservation such as for energy or momentum, initially established for material points, will also be valid for electromagnetic fields if the Maxwell equations can be cast in Lagrangian form. And indeed they can.

The implications are actually very far reaching and have repercussions in quantum mechanics and in quantum field theory through Noether's theorem relating the invariance of the Lagrangian to certain symmetry transformations and the establishment of conservation laws.

Thus, if a differential equation is written with variables that have no direct relations with the positions of a particle system but deriving from a Lagrangian that can take the form of a particle system invariant to translation in time, then a function $H$ that corresponds to an energy and that will satisfy a conservation law can be defined.

The Lagrange equations are differentiable provided the function $S$

$S=\int_{t1}^{t2} L (q_{i}, \dot{q_{i}})dt$

Is an extremum for all the variations (in a precise direction) of the coordinates of position and velocity of a mechanical system having a Lagrangian between two dates separating the movement states of the mechanical system being considered.

This function $S$ is called the action, and is used to write the following equations:

$H(q_i,p_i)=-\frac {\partial S}{\partial t}$

$p_i=\frac{\partial S}{\partial q_i}$

The first is precisely the Hamilton-Jacobi equation and is connected to the integration of the preceding Hamilton equations.

It is this equation that allowed Schrödinger to develop wave mechanics.

The Lagrangian of a mechanical system is the basis of the formulation of quantum mechanics using Richard Feynman's famous path integral. The whole of particle physics in the standard model along with the quantum theory of relativistic fields is formulated with the help of a Lagrangian.

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It is this equation that allowed Schrödinger to develop wave mechanics.

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