Why is the Lagrangian density defined in fields

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The Lagrangian density \ ({\ displaystyle {\ mathcal {L}}} \) (after the mathematician Joseph-Louis Lagrange) plays a role in theoretical physics when considering fields. It describes the density of the Lagrange function \ ({\ displaystyle L} \) in a volume element. Therefore the Lagrange function is defined as the integral of the Lagrange density over the volume under consideration:

\ ({\ displaystyle L = \ int \ mathrm {d} ^ {3} r {\ mathcal {L}} = \ iiint \ mathrm {d} x \, \ mathrm {d} y \, \ mathrm {d} z \, {\ mathcal {L}} \ left (\ phi, {\ frac {\ partial \ phi} {\ partial t}}, {\ frac {\ partial \ phi} {\ partial x}}, {\ frac {\ partial \ phi} {\ partial y}}, {\ frac {\ partial \ phi} {\ partial z}}, t \ right)} \)

with the considered field \ ({\ displaystyle \ phi (x, y, z, t)} \).

The real purpose of the Lagrange density is to describe fields using equations of motion. Just as the Lagrange equations of the second kind are obtained from the Hamiltonian principle, the Lagrange equations for fields can be obtained from the Hamiltonian principle for fields (derivation). Accordingly, the equation of motion reads:

\ ({\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi _ {i}}} - {\ frac {\ mathrm {d}} {\ mathrm {d} t}} {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi _ {i}} {\ partial t}}}} - \ sum _ {j = 1} ^ {3 } {\ frac {\ mathrm {d}} {\ mathrm {d} x_ {j}}} {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi _ { i}} {\ partial x_ {j}}}}} = {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi _ {i}}} - \ partial _ {\ mu} {\ frac {\ partial {\ mathcal {L}}} {\ partial (\ partial _ {\ mu} \ phi _ {i})}} = 0} \).

example


For a string vibrating in one dimension, the result is Lagrangian density

\ ({\ displaystyle {\ mathcal {L}} = {\ frac {1} {2}} \ left [\ mu \ left ({\ frac {\ partial \ phi} {\ partial t}} \ right) ^ {2} -E \ left ({\ frac {\ partial \ phi} {\ partial x}} \ right) ^ {2} \ right]} \)

In this example:

\ ({\ displaystyle \ phi = \ phi (x, t)} \) the deflection of a point of the string from the rest position (field variable)
\ ({\ displaystyle \ mu} \) the linear mass density
\ ({\ displaystyle E} \) the modulus of elasticity

With this Lagrangian density surrendered

\ ({\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi}} = 0} \)
\ ({\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial t}}}} = \ mu {\ frac {\ partial \ phi } {\ partial t}}} \)
\ ({\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ frac {\ partial \ phi} {\ partial x}}}} = - E {\ frac {\ partial \ phi } {\ partial x}}} \)

This results in the Equation of motion the vibrating string

\ ({\ displaystyle E {\ frac {\ partial ^ {2} \ phi} {\ partial x ^ {2}}} - \ mu {\ frac {\ partial ^ {2} \ phi} {\ partial t ^ {2}}} = 0} \)

Application in the theory of relativity


The description of physical processes via the Lagrange density is used instead of the Lagrange function, especially in relativistic processes. Here a covariant representation of the Lagrange function is desired, then the effect is over

\ ({\ displaystyle S = \ int \ mathrm {d} ^ {4} x \, {\ mathcal {L}}} \)

Are defined. The Lagrange function is thus a Lorentz scalar, i.e. invariant under Lorentz transformations:

\ ({\ displaystyle {\ mathcal {L}} '(x _ {\ mu}) = {\ mathcal {L}} (x' _ {\ mu}) = {\ mathcal {L}} (x _ {\ mu })} \) with \ ({\ displaystyle x '_ {\ mu} = \ Lambda _ {\ mu \ nu} x ^ {\ nu}} \), where \ ({\ displaystyle \ Lambda _ {\ mu \ nu}} \) is the Lorentz transformation tensor.

literature


  • Franz Schwabl: Lagrangian density. In the S.: Quantum Mechanics for Advanced Students (QM II). Springer, Berlin 2005, ISBN 978-3-540-28865-7, pp. 281ff.









Categories:Field theory | Classic mechanics | Joseph-Louis Lagrange as namesake




Status of information: 11/24/2020 9:54:17 AM CET

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