Let be a mechanical system with the configuration space and the smooth Lagrangian Select a standard … See more

Hamilton's equations can be derived by a calculation with the Lagrangian , generalized positions q , and generalized velocities , where . Here we work off-shell, meaning , , are independent coordinates in phase space, not … See more

• The value of the Hamiltonian is the total energy of the system if and only if the energy function has the same property. (See definition of ).
• when , form a solution of Hamilton's equations. Indeed, and everything … See more

In its application to a given system, the Hamiltonian is often taken to be
where is the kinetic energy and is the potential energy. Using this relation can be simpler than first calculating the Lagrangian… See more

A sufficient illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an electromagnetic field. In Cartesian coordinates the Lagrangian of a non-relativistic classical particle in an electromag… See more

The Hamiltonian can induce a symplectic structure on a smooth even-dimensional manifold M in several equivalent ways, the best known being the following:
As a closed nondegenerate symplectic 2-form ω. Accordi… See more

• Canonical transformation
• Classical field theory
• Hamiltonian field theory
• Covariant Hamiltonian field theory… See more