Hamiltonian mechanics | Symplectic geometry

Hamiltonian vector field

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics. Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f and g on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by thePoisson bracket of f and g. (Wikipedia).

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The gradient of a function is a vector. n-Dimensional space can be filled up with countless vectors as values as inserted into a gradient function. This is then referred to as a vector field. Some vector fields have potential functions. In this video we start to look at how to calculat

From playlist Advanced Calculus / Multivariable Calculus

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Introduction to Vector Fields This video discusses, 1) The definition of a vector field. 2) Examples of vector fields including the gradient, and various velocity fields. 3) The definition of a conservative vector field. 4) The definition of a potential function. 5) Test for conservative

From playlist Calculus 3

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From playlist Quantum Field Theory

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From playlist Multivariable Calculus

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From playlist Integration and Vector Fields

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From playlist Calculus 3 (Full Length Videos)

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From playlist Multivariable Calculus

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From playlist THE "WHAT IS" PLAYLIST

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From playlist CALCULUS 3 CH 7 GREEN'S THEOREM

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From playlist Virtual Conference

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From playlist 8.05 Quantum Physics II - Prof. Barton Zwiebach

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From playlist MIT 8.422 Atomic and Optical Physics II, Spring 2013

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From playlist Mathematics

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From playlist Mathematics

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From playlist Mathematics

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From playlist Virtual Conference

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From playlist 8.05 Quantum Physics II - Prof. Barton Zwiebach

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From playlist 8.05 Quantum Physics II - Prof. Barton Zwiebach

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From playlist Course | Quantum Entanglements: Part 1 (Fall 2006)

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From playlist Calculus IV: Vector Calculus (Line Integrals, Surface Integrals, Vector Fields, Greens' Thm, Divergence Thm, Stokes Thm, etc) **Full Course**

Related pages

Tangent bundle | Exterior derivative | Hamiltonian mechanics | Lie bracket of vector fields | Noether's theorem | Kernel (linear algebra) | Poisson manifold | Differentiable function | Isomorphism | Symplectomorphism | Symplectic vector space | William Rowan Hamilton | Poisson bracket | Cotangent bundle | Linear map | Mathematics | Lie derivative | Diffeomorphism | Lie algebra | Canonical transformation | Integral curve | Bilinear form | Flow (mathematics) | Symplectic manifold | Canonical coordinates | Conservation of energy | Jacobi identity | Exterior product | Vector field