**Elastic Energy**Closed Elastic Curves and Closed Elastic Curves and Rods Rods III Kirchhoff Rods III Kirchhoff Rods Kirchhoff Elastic Rod The Darboux Vector Inertial Frames 1Torsion and Twisting 2

Integrating the rod equations Closed Elastic Rods Closed Elastic Curves and Closed Elastic Curves and Rods Rods IV Hamiltonian systems IV Hamiltonian systems Geometric Hamiltonian Systems 3

Poisson Brackets 4

Lie Bracket Formulas Hamiltonians on E Symmetries simplify! 5

Frenet Equations Example: The Kirchhoff Rod Pontrjagin Maximum Principle Liouville –Arnol’d Integrability 6

End of Part 4 End of Part 4 More Examples ? 7