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Lagrange equations

The Lagrange equations are the fundamental equations of analytical mechanics. There is another more powerful formulation of analytical mechanics called the Hamiltonian formulation, but this is naturally tied to the Lagrange formulation. Both are pillars of theoretical physics and their applications in all domains of physics are numerous.

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

The Lagrangian formulation is at the heart of relativistic quantum field theory in which it is indispensable for quantifying the gauge theories of Yang-Mills. It also allows a flexible formulation of the theory of elementary particles and string theory.

In this respect the Lagrange formulation of quantum theory in the form of a Feynman path integral is currently the most powerful tool. In particular, it can be used for problems such as the evaporation of black holes, and quantum cosmology problems such as Stephen Hawking and James Hartle have shown.

The Lagrange equations actually have several origins, and result from work by Descartes, d'Alembert and Euler in particular.

Newtonian mechanics and Newton's and Leibnitz's infinitesimal calculus were applied in the 18th century to tackle all kinds of problem in celestial mechanics and the mechanics of solids. The proliferation of differential equations and special methods of solution was becoming a concern and calculations were becoming increasingly lengthy and difficult. An overall unifying vision was lacking.

D'Alembert had shown that problems in dynamics could be reduced to problems in statics and Euler, basing his work on that of the Bernoulli brothers, had introduced a general calculus to determine the extrema, not of functions of numbers, but of functions of functions, which were subsequently called functionals. Two famous examples were the isoperimetric problem and the brachistochrone problem.

Leonhard Euler and his famous identity (15 April 1707 - 18 September 1783)

In the first case the problem was to find the maximum surface area for a given perimeter (it can for example be proved that for a given perimeter, a square contains a greater area than a rectangle). In the second case the problem was to find the curve of fastest descent for a body acted on by gravity. It is in fact a cycloid.

Lagrange was thus trying to do for mechanics what Descartes had done for geometry when he created analytical geometry. He gave a general formulation of the equations of mechanics which, like the equations of lines and planes in the geometry of Descartes, unified many specific mechanics problems. It also unified the methods for the solution of these problems and made it possible to demonstrate theorems in mechanics in a way that was automatically valid for large classes of mechanical systems.

The final result was the reduction of mechanics to analytical geometry in close connection with differential and integral calculus. Moreover, the equations of motion of any system could be derived from a so-called variational principle based on the determination of the extrema of a functional.

A simple example will suffice to provide concrete understanding of the meaning of these equations.

Consider a collection of N particles such as a gas or the planets around the Sun. They therefore have 3N position coordinates $\dot{q_{i}}$ and 3N velocity coordinates $\dot{q_{i}}$. The dot denotes the total derivative of the preceding coordinate with respect to time, i.e. $\dot{q_{i}}=\frac{dq_{i}}{dt}$.

These coordinates are referred to as generalised because it is unnecessary to specify whether Cartesian, spherical or other coordinates are being used. For the moment we will assume them to be Cartesian.

Lagrange then showed that all the second order differential equations of motion with respect to time that were known in mechanics could be expressed in the form:

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

where the function $L$ is called a Lagrange function or the Lagrangian. For the above particles in a potential V it 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.

More generally, for N particles in an arbitrary generalised coordinate system we will have, in matrix notation:

$L( q_i,\dot{q_i})=\sum_{jk}^{3n}A _{jk}(q_i) \dot{q^j} \dot{q^k}- V(q_i)$

The Lagrangian formulation of equations is, however, invariant to a change in the coordinate system $q_{i}$ and this is what makes it so powerful because large classes of differential equations expressed in Cartesian, spherical etc. coordinates can be reduced to a few fundamental cases that we know how to solve. There is thus a powerful technique for changing variables in differential and integral calculus.

Furthermore, 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 equations in Lagrangian form.

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 relationship to 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 $H$ can be defined that corresponds to an energy and that will satisfy a conservation law.

There is one point that it is very important to understand. The Lagrange equations were first obtained for a system of material points, but the general nature of the equations obtained by using an arbitrary coordinate system means that differential equations of change for different physical systems, such as electrical currents in oscillating circuits or animal populations in an ecosystem, can have the same form as for a mechanical system of points. This is why a mechanical system and more generally a dynamic system, is referred to every time (at least, but not exclusively) a system of second order differential equations with respect to time can be cast in Lagrangian form.

Finally, on the basis of the work of Maupertuis, whose work was itself based on one of Fermat' ideas, Lagrange and Hamilton showed that the Lagrange equations are differentiable on condition that the function $S$ defined by:

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

is an extremum for all the variations (in a precise sense) of the coordinates of position and velocity of a mechanical system endowed with a Lagrangian, between two dates separating the movement states of the mechanical system being considered.

This function $S$ is called the action.

It is this that makes it possible to define the well known principle of least action and 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 it is connected to the integration of the preceding Hamilton equations.

It was using this equation that Schrödinger developed his wave mechanics from the ideas of De Broglie and the remarks of Félix Klein (Not the Oscar of Kaluza-Klein).

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