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:
“The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we don’t like what we see, we feel angry at the reflection.”
—Elaine Heffner (20th century)
“Culture is a sham if it is only a sort of Gothic front put on an iron building—like Tower Bridge—or a classical front put on a steel frame—like the Daily Telegraph building in Fleet Street. Culture, if it is to be a real thing and a holy thing, must be the product of what we actually do for a living—not something added, like sugar on a pill.”
—Eric Gill (1882–1940)
“In the quilts I had found good objects—hospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.”
—Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)
“Many people have an oversimplified picture of bonding that could be called the “epoxy” theory of relationships...if you don’t get properly “glued” to your babies at exactly the right time, which only occurs very soon after birth, then you will have missed your chance.”
—Pamela Patrick Novotny (20th century)