Extensions To Classical Field Theory
- Lagrangian field theory
Replacing the generalized coordinates by scalar fields φ(r, t), and introducing the Lagrangian density (Lagrangian per unit volume), in which the Lagrangian is the volume integral of it:
where ∂μ denotes the 4-gradient, the Euler-Lagrange equations can be extended to classical fields (such as Newtonian gravity and classical electromagnetism):
where the summation convention has been used. This formulation is an important basis for quantum field theory - by replacing wavefunctions with scalar fields.
- Hamiltonian field theory
The corresponding momentum field density conjugate to the field φ(r, t) is:
The Hamiltonian density (Hamiltonian per unit volume) is likewise;
and satisfies analogously:
Read more about this topic: Analytical Mechanics
Famous quotes containing the words extensions, classical, field and/or theory:
“If we focus exclusively on teaching our children to read, write, spell, and count in their first years of life, we turn our homes into extensions of school and turn bringing up a child into an exercise in curriculum development. We should be parents first and teachers of academic skills second.”
—Neil Kurshan (20th century)
“Several classical sayings that one likes to repeat had quite a different meaning from the ones later times attributed to them.”
—Johann Wolfgang Von Goethe (1749–1832)
“The birds their quire apply; airs, vernal airs,
Breathing the smell of field and grove, attune
The trembling leaves, while universal Pan,
Knit with the Graces and the Hours in dance,
Led on th’ eternal Spring.”
—John Milton (1608–1674)
“There is in him, hidden deep-down, a great instinctive artist, and hence the makings of an aristocrat. In his muddled way, held back by the manacles of his race and time, and his steps made uncertain by a guiding theory which too often eludes his own comprehension, he yet manages to produce works of unquestionable beauty and authority, and to interpret life in a manner that is poignant and illuminating.”
—H.L. (Henry Lewis)