Hamiltonian

Generalized Momentum  The momentum can be expressed in terms of the kinetic energy.  A generalized momentum can be defined similarly. Kinetic energy in generalized coordinatesKinetic energy in generalized coordinates Equivalent to transformed Newtonian momentumEquivalent to transformed Newtonian momentum

Momentum in Lagrangian  The momentum can be derived from the Lagrangian. Potential energy independent of velocity  The momentum can be used in the EL equations.

Ignorable Coordinates  If the Lagrangian doesn’t depend on a generalized coordinate, the generalized momentum is conserved. assume then

Projectile Motion  The Lagrangian for a projectile does not depend on x. x -momentum conservedx -momentum conserved x y m

Eliminating Coordinates  Ignorable (cyclic) coordinates can be eliminated from a Lagrangian. Defines the Routhian Reduced degrees of freedom by 1  The new Lagrangian gives the same equations of motion.

Hamiltonian  The idea of the Routhian can be extended. Include all coordinatesInclude all coordinates Defines the HamiltonianDefines the Hamiltonian  The time derivative of H depends on the individual coordinates and velocities.

Energy Conservation Use  The EL equations can be used to eliminate terms from the time derivative.  If the Lagrangian is independent of time, the Hamiltonian is conserved. Expressed as total energy next