Lagrangian density

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

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

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

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 Hamilton's principle, the Lagrange equations for fields can be obtained from Hamilton's 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

Exemplary solution of the equation of motion of a vibrating string in 3 dimensions. Parameters:, animation runs at 10% of the actual speed. ${\displaystyle E=\mu =1}$

For a string vibrating in one dimension , the Lagrange density results

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

In this example:

${\displaystyle \phi =\phi (x,t)}$ the deflection of a point on the string from its rest position (field variable)

${\displaystyle \mu }$ the linear mass density

${\displaystyle E}$the modulus of elasticity

With this Lagrange density we get

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \phi }}=0}$

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial t}}}}=\mu {\frac {\partial \phi }{\partial t}}}$

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial x}}}}=-E{\frac {\partial \phi }{\partial x}}}$

This results in the equation of motion for 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 therefore a Lorentz scalar, i.e. invariant under Lorentz transformations :

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

literature

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