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# 8: Hamiltonian Mechanics

• 8.1: Introduction
Hamiltonian mechanics plays a fundamental role in modern physics.
• 8.2: Legendre Transformation between Lagrangian and Hamiltonian mechanics
Hamiltonian mechanics can be derived directly from Lagrange mechanics by considering the Legendre transformation between the conjugate variables (q,q˙,t) and (q,p,t) . Such a derivation is of considerable importance in that it shows that Hamiltonian mechanics is based on the same variational principles as those used to derive Lagrangian mechanics; that is d’Alembert’s Principle and Hamilton’s Principle.
• 8.3: Hamilton’s Equations of Motion
Canonical equations of motion.
• 8.4: Hamiltonian in Different Coordinate Systems
Prior to solving problems using Hamiltonian mechanics, it is useful to express the Hamiltonian in cylindrical and spherical coordinates for the special case of conservative forces since these are encountered frequently in physics.
• 8.5: Applications of Hamiltonian Dynamics
The equations of motion of a system can be derived using the Hamiltonian coupled with Hamilton’s equations of motion.
• 8.6: Routhian Reduction
It is advantageous to have the ability to exploit both the Lagrangian & Hamiltonian formulations simultaneously for systems that involve a mixture of cyclic and non-cyclic coordinates. The equations of motion for each independent generalized coordinate can be derived independently of the remaining generalized coordinates. Thus it is possible to select either Hamiltonian or Lagrangian formulations for each generalized coordinate, independent of what is used for the other generalized coordinates.
• 8.7: Variable-mass systems
Lagrangian & Hamiltonian mechanics assume that the total mass and energy of the system are conserved. Variable-mass systems involve transferring mass and energy between donor and receptor bodies. However, such systems still can be conservative if the Lagrangian or Hamiltonian include all the active degrees of freedom for the combined donor-receptor system. The following examples of variable mass systems illustrate subtle complications that occur handling such problems using algebraic mechanics.
• 8.E: Hamiltonian Mechanics (Exercises)
• 8.S: Hamiltonian Mechanics (Summary)