We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model Lagrangian density.
12.
Lagrangian mechanics is widely used to solve mechanical problems in physics and engineering when fields are described using a Lagrangian density.
13.
The Lagrangian densities of the theories on the lower-dimensional branes may be obtained from holomorphic Chern Simons theory by dimensional reductions.
14.
Starting with a non-symmetric tensor g _ { \ mu \ nu } \;, the Lagrangian density is split into
15.
If a Lagrangian density ( including interactions ) is available, then the Lagrangian formalism will yield an equation of motion at the classical level.
16.
The path-integral formulation provides the most direct way from the Lagrangian density to the corresponding Feynman series in its Lorentz-invariant form.
17.
The Hamiltonian density is related to the Lie derivative of the Lagrangian density with respect to a unit timelike horizontal vector field on the gauge bundle.
18.
The Lagrangian density is a function of the fields in the system, their space and time derivatives, and possibly the space and time coordinates themselves.
19.
The dynamics of the quantum state and the fundamental fields are determined by the Lagrangian density \ mathcal { L } ( usually for short just called the Lagrangian ).
20.
That a boundary term is needed in the gravitational case is due to the fact that R, the gravitational Lagrangian density, contains second derivatives of the metric tensor.
