Continuous Systems and Fields

13.1 Continuous systems So far all the formulations have been devised for treating systems with a discrete number of degrees of freedom There are many problems, mechanical in particular, that involve continuous description E.g., elastic solid: each point of the continuous solid participates in the motion, which can only be described by specifying the position coordinates of all points We have to modify the previous formulations to handle such problems using concepts of field theory

Continuous systems Recipe: 13.1 Continuous systems Recipe: a) approximate the continuous system with a discrete system b) solve that problem c) take continuous limit

13.1 Example Infinitely long elastic rod that can undergo small longitudinal vibrations Approximation – an infinite chain of equal mass points spaced distance a apart and connected by uniform massless springs Mass points can move along the length of the chain and displacements from equilibrium are small

Elastic rod: Lagrangian formulation 13.1 Elastic rod: Lagrangian formulation Kinetic energy: Potential energy: Lagrangian:

Elastic rod: equations of motion 13.1 Elastic rod: equations of motion Euler-Lagrange equations: Linear density

Elastic rod: equations of motion 13.1 Elastic rod: equations of motion The rod is elastic and obeys Hooke’s law: Y – Young’s modulus The force necessary to stretch i-th spring: Thomas Young (1773 – 1829)

Elastic rod: equations of motion 13.1 Elastic rod: equations of motion Going from the discrete to the continuous: Wave equation

Elastic rod: equations of motion 13.1 Elastic rod: equations of motion Going from the discrete to the continuous: Back to Lagrangian:

Elastic rod: equations of motion 13.1 Elastic rod: equations of motion Lagrangian density:

Lagrangian formulation 13.2 Lagrangian formulation General 1D case: Action: Variational principle:

Lagrangian formulation 13.2 Lagrangian formulation

Lagrangian formulation 13.2 Lagrangian formulation General 3D case: Integrating by parts:

Lagrangian formulation 13.2 Lagrangian formulation Field – an independent function of space and time There is no requirement that the field be related to some mechanical system, (e.g. electromagnetic field) Lagrangian density also contains information on the conserved properties of the system

Conservation laws Calculating the full derivative: 13.3 Conservation laws Calculating the full derivative: From the equations of motion: Combining total derivatives:

13.3 Conservation laws If the Lagrangian density does not depend explicitly on xμ this usually means that η represents a free field (i.e. it contains no external driving sources or sinks interacting with the field at explicit space points and with given time dependence) Equations of continuity

Hamiltonian formulation 13.4 Hamiltonian formulation Hamiltonian formulation can be introduced in a straightforward manner for classical fields This procedure singles out the time variable for special treatment in contrast to the Lagrangian formulation where the independent variables of time and space are handled symmetrically Thus, the Hamiltonian approach for fields has not proved as useful as the Lagrangian method