How Do Poisson Brackets Function in Classical Mechanics?

In summary, Poisson brackets are a fundamental tool in the Hamiltonian formulation of classical mechanics, used to express equations of motion and constraints on changed canonical variables. They are also related to commutators of operators in quantum mechanics. The Poisson bracket is defined for functions f and g as a specific sum involving partial derivatives of f and g with respect to canonical variables (q,p). The equation of motion for quantity f is then given by adding the partial derivative of f with respect to time to the Poisson bracket of f with the Hamiltonian. A change of variables from canonical variables (q,p) to canonical variables (Q,P) has specific constraints on their Poisson brackets. The proof of the equation of motion involves using Hamilton's equations
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Definition/Summary

In the Hamiltonian formulation of classical mechanics, equations of motion can be expressed very conveniently using Poisson brackets. They are also useful for expressing constraints on changed canonical variables.

They are also related to commutators of operators in quantum mechanics.

Equations

For canonical variables (q,p), the Poisson bracket is defined for functions f and g as
[itex]\{f,g\} = \sum_a \left( \frac{\partial f}{\partial q_a}\frac{\partial g}{\partial p_a} - \frac{\partial f}{\partial p_a}\frac{\partial g}{\partial q_a} \right)[/itex]

The equation of motion for quantity f is
[itex]\dot f = \frac{\partial f}{\partial t} + \{f,H\}[/itex]

A change of variables from canonical variables (q,p) to canonical variables (Q,P) has these constraints:
[itex]\{Q_i,P_j\} = \delta_{ij} ,\{Q_i,Q_j\} = \{P_i,P_j\} = 0[/itex]

Extended explanation

Proof of equation of motion.

Start with
[itex]\frac{df}{dt} = \frac{\partial f}{\partial t} + \sum_a \left( \frac{\partial f}{\partial q_a} \frac{dq_a}{dt} + \frac{\partial f}{\partial p_a} \frac{dp_a}{dt} \right)[/itex]

Using Hamilton's equations of motion gives
[itex]\frac{df}{dt} = \frac{\partial f}{\partial t} + \sum_a \left( \frac{\partial f}{\partial q_a} \frac{\partial H}{\partial p_a} - \frac{\partial f}{\partial p_a} \frac{\partial H}{\partial q_a} \right) = \frac{\partial f}{\partial t} + \{f,H\}[/itex]

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Thanks for the overview of Poisson brackets
 

Related to How Do Poisson Brackets Function in Classical Mechanics?

What is a Poisson bracket?

A Poisson bracket is a mathematical operation used in classical mechanics to describe the relationship between two physical quantities, such as position and momentum. It is represented by curly brackets and is used to calculate the rate of change of one quantity with respect to the other.

How is a Poisson bracket calculated?

A Poisson bracket is calculated by taking the partial derivatives of the two quantities involved and then multiplying them together, before taking the difference between the two products.

What is the significance of a Poisson bracket?

A Poisson bracket is significant because it allows us to describe the dynamics of a physical system and make predictions about its future behavior. It is also a fundamental concept in the mathematical formulation of classical mechanics.

Can a Poisson bracket be used in other fields besides classical mechanics?

Yes, the concept of a Poisson bracket can also be applied in other fields such as quantum mechanics, differential geometry, and even economics. It is a versatile tool in mathematics and can be used to study a wide range of systems and phenomena.

Who first introduced the concept of a Poisson bracket?

The concept of a Poisson bracket was first introduced by French mathematician and physicist Siméon Denis Poisson in the early 19th century. It was originally used to describe the motion of celestial bodies, but has since been applied to various other fields of study.

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