Advanced Classical Mechanics/Hamilton's Equations

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Hamilton's Equations[edit]

Hamilton's equations are an alternative to the Euler-Lagrange equations to find the equations of motion of a system. They are applied to the Hamiltonian function of the system. Hamilton's equations are

Where is the Hamiltonian, are the generalized coordinates, are the generalized momenta, and a dot represents the total time derivative. The Hamiltonian can be found by performing a Legendre transformation on the Lagrangian of the system:

where the Einstein summation notation is used, a dot represents the total time derivative, qi are the generalized coordinates, pi are the generalized momenta.

These momenta are found by differentiating the Lagrangian with respect to the generalized velocities . Mathematically


In some cases, is the total energy of the system. There are two conditions for this. First, the equations defining the generalized coordinates q don't depend on time explicitly. Second, the forces involved in the system are derivable from a scalar potential. The forces must be conservative (such as gravity).[1]


  1. Goldstein, Poole, Safko. Classical Mechanics 3rd ed. 2002