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)
“I don’t like comparisons with football. Baseball is an entirely different game. You can watch a tight, well-played football game, but it isn’t exciting if half the stadium is empty. The violence on the field must bounce off a lot of people. But you can go to a ball park on a quiet Tuesday afternoon with only a few thousand people in the place and thoroughly enjoy a one-sided game. Baseball has an aesthetic, intellectual appeal found in no other team sport.”
—Bowie Kuhn (b. 1926)
“No theory is good unless it permits, not rest, but the greatest work. No theory is good except on condition that one use it to go on beyond.”
—André Gide (1869–1951)