CAARMS11 Tutorial on Hamiltonians

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11th Conference for African American Researchers in the Mathematical Sciences Institute for Pure and Applied Mathematics, June 21-24, 2005 Tutorial on the Calculus of Variations:Part II, HamiltoniansWilliam A Massey,Princeton Universitywmassey@princeton.edu1CAARMS ACKNOWLEDGEMENTSThanks to Prof. Rudy L. Horne for pointing out to me the connections between optimization and Hamiltonian systems.These notes also benefited from many discussions withProf. Earl Barnes.2££”Application Interpretation for Hamiltonians: Classical MechanicsThe curve x x(t) 0 t T is the evolution of{ }the position of a system and the Hamiltonian H is theenergy of the system. A least action principle for a Hamiltonian with no explicit time dependence yields a conservation of energyprinciple. Similary, a Hamiltonian with no explicit position dependence yields a conservation of momentum principle.3Classical Mechanics Example:The Two Body Problem4”Extremal Notationext min or max5”-Legendre Transforms( )h(y) ext x? y x( )x6Convex function ℓ(x,xy)(x,0)(x,ℓ(x))7SlopeyMaximum Difference xy-ℓ(x)--Legendre Transform Example21 y2(x) = m ?x ? h(y)=2 2m2 21 y 1 y y 2h(y) = max x ?y m? x = ?y m? = 2x 2  m 2 m 2m8ﬁ--Fundamental Result forLegendre TransformsIf : is a nice function, then so is h and( )h(y) = ext x ?y x( )x(x) = ext x ?y h y( )( )yWhen we have min or max, then strict ...

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