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This lesson covers the fundamentals of Hamiltonians in physics, exploring their role in mechanical systems. It explains how the Hamiltonian ( H ) is formulated as ( H = T + U ) and its relation to the Lagrangian method, which yields equations of motion. The session discusses the significance of generalized momentum and the conservation of ( H ) under conditions of explicit time dependence. The interplay between the Lagrangian and the Hamiltonian is emphasized, highlighting that one often must derive the Lagrangian to determine the Hamiltonian effectively.
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Physics 321 Hour 20 Hamiltonians
whereas • For simple systems, H = T + U • There are two first order equations of motion for each variable: • The Lagrangian method gives one second order equation for each variable. The Hamiltonian
Take Therefore The Hamiltonian - Origins
Let If has no explicit time dependence, then H is a conserved quantity. The Hamiltonian - Origins
The Hamiltonian is a function of p and q. But p is not ‘the momentum,’ it is the generalized momentum conjugate to q. • The general expression is: • That means you generally have to find the Lagrangian before you can find he Hamiltonian! The Hamiltonian - Notes
The Hamiltonian is H=T+U unless • The Lagrangian has explicit time dependence • The transormation between the coordinatiates and Cartesian coordinates have explicit time dependence • But … you have to write the Hamiltonian in terms of the correct generalized momenta – so you usually still need the Lagrangianfirst! The Hamiltonian - Notes
hamilton.nb Example
