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The Lagrange formulation

Importance

The Lagrange formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics, the action principle that is used to derive the Lagrange equation is now recognized to be applicable to quantum mechanics.

Physical action and quantum-mechanical phase are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions.

The same principle, and the Lagrange formalism, are tied closely to Noether's theorem, which relates physical conserved quantities to continuous symmetries of a physical system.

Lagrangian mechanics and Noether's theorem together yield a natural formalism for first quantization by including commutators between certain terms of the Lagrangian equations of motion for a physical system.

Advantages over other methods

• The formulation is not tied to any one coordinate system -- rather, any convenient variables may be used to describe the system; these variables are called "generalized coordinates" and may be any independent variable of the system (for example, strength of the magnetic field at a particular location; angle of a pulley; position of a particle in space; or degree of excitation of a particular eigenmode in a complex system). This makes it easy to incorporate constraints into a theory by defining coordinates which only describe states of the system which satisfy the constraints.

• If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This is very helpful in showing that theories are consistent with either special relativity or general relativity.

• Equations derived from a Lagrangian will almost automatically be unambiguous and consistent, unlike equations brought together from multiple formulations.

"Cyclic coordinates" and conservation laws

An important property of the Lagrangian is that conservation laws can easily be read off from it. E.g., if the Lagrangian depends on the time-derivative of a generalized coordinate, but not on q_i itself, then the generalized momentum,

,

is a conserved quantity. This is a special case of Noether's theorem, see below. Such coordinates are called "cyclic".

For example, the conservation of the generalized momentum,

,

say, can be directly seen if the Lagrangian of the system is of the form

Also, if the time, t, does not appear in , then the conservation of the Hamiltonian follows. This is the energy conservation unless the potential energy depends on velocity, as in electrodynamics. More details can be found in any textbook on theoretical mechanics.

Source : Wikipedia