1. Continuous Systems and Fields

2. 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

3. 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

4. 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

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

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

7. 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)

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

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

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

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

12. Lagrangian formulation
13.2
Lagrangian formulation

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

14. 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

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

16. 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

17. 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