How did the Poisson brackets get derived, and from what.

Thread starter houlahound
Start date Apr 18, 2015
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Poisson Poisson brackets

In summary, the Poisson brackets were derived by French mathematician Siméon Denis Poisson in the early 19th century as a way to describe the dynamical behavior of systems with many degrees of freedom. Poisson was inspired by the Hamiltonian formalism of classical mechanics and used the concept of the Lie bracket in differential geometry to create a more concise and elegant form of the Hamiltonian equations. The Poisson brackets are closely related to the commutator and the Poisson bracket, and have practical applications in physics, engineering, chaos theory, and control theory.

Apr 18, 2015

houlahound

do they have a physical meaning and did they fall out of another theory.

I have only ever seen them stated as a fact, I am assuming they are a result of something ie a consequence of another more fundamental theory.

when are they used in a practical problem solving sense to solve real world problems?

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Apr 18, 2015

kreil

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http://en.wikipedia.org/wiki/Poisson_bracket

FAQ: How did the Poisson brackets get derived, and from what.

1. How did the Poisson brackets get derived?

The Poisson brackets were first derived by French mathematician Siméon Denis Poisson in the early 19th century. He was studying the mathematical properties of celestial mechanics and needed a way to describe the dynamical behavior of systems with many degrees of freedom.

2. What was the inspiration for the Poisson brackets?

Poisson was inspired by the Hamiltonian formalism of classical mechanics, which uses a set of equations known as Hamilton's equations to describe the evolution of a physical system over time. These equations involve a quantity called the Hamiltonian, which represents the total energy of the system. Poisson wanted to find a way to express the Hamiltonian equations in a more concise and elegant form.

3. From what mathematical concepts were the Poisson brackets derived?

The Poisson brackets were derived from the concept of the Lie bracket in differential geometry. This is a mathematical operation that describes how two vector fields change as they are composed together. Poisson realized that this operation could be applied to the Hamiltonian equations to create a new set of equations that were more compact and easier to work with.

4. How do the Poisson brackets relate to other mathematical concepts?

The Poisson brackets are closely related to two other mathematical concepts: the commutator and the Poisson bracket. The commutator is a mathematical operation that describes how two operators behave when they are applied in different orders. The Poisson bracket is a generalization of the commutator that applies to continuous functions instead of operators. The Poisson brackets are essentially a continuous version of the commutator.

5. What are some practical applications of the Poisson brackets?

The Poisson brackets are used in many areas of physics and engineering, including classical mechanics, quantum mechanics, and thermodynamics. They are particularly useful for studying systems with many degrees of freedom, such as fluids and plasmas. They are also used in the study of chaos and nonlinear dynamics, where they provide insights into the behavior of complex systems. Additionally, the Poisson brackets have applications in control theory, where they are used to analyze the stability and controllability of systems.