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
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