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)
“Classical art, in a word, stands for form; romantic art for content. The romantic artist expects people to ask, What has he got to say? The classical artist expects them to ask, How does he say it?”
—R.G. (Robin George)
“The planter, who is Man sent out into the field to gather food, is seldom cheered by any idea of the true dignity of his ministry. He sees his bushel and his cart, and nothing beyond, and sinks into the farmer, instead of Man on the farm.”
—Ralph Waldo Emerson (1803–1882)
“Could Shakespeare give a theory of Shakespeare?”
—Ralph Waldo Emerson (1803–1882)