Lagrangian for a Long Elastic Rod

1. Equilibrium Positions a Lagrangian for a Long Elastic Rod Approximate by: Discrete masses m at equilibrium points a. Kinetic Energy Potential Energy

2. Lagrangian for a Long Elastic Rod Discrete Approximation Lagrangian where Liis a Lagrangian for each little piece Equations of motions:

3. Limit as a→0 Note that x and t are index variables! Y=Young’s modulus

4. Hamilton’s Principle where L is the Lagrangian density Example – Elastic Rod Parameterize different paths as: Hamiltonian’s principle: is an extreme for all

5. Hamilton’s Principle to Equations of Motion Equation of Motion: Integration by parts – Surface term is zero because endpoints are fixed.

6. Elastic Rod Example Wave equation with Discrete equation of motion a→0

7. Four Dimensional Notation Four dimensional notation: Equation of Motion is invariant Since For fields with more than one degree of freedom (vectors, etc.) For each DOF.

8. Stress Energy Tensor

9. Stress Energy Tensor (continued) Stress-Energy Tensor Example: Hamiltonian density (i.e., Lagrangian density does not explicitly depend on position), then If: Leads to conserved currents

10. Conserved Currents Current If volume is big enough such that there is no flow out. In this case, Rm is conserved (i.e. doesn’t change in time). R0= total energy Ri = total momentum in ith direction