Glossary

6.2 Poisson brackets and canonical invariants

2 min read•july 25, 2024

Poisson brackets are a powerful tool in classical mechanics, quantifying how variables change relative to each other in phase space. They have key properties like and , and form the basis for understanding and in physical systems.

Time evolution in mechanics can be elegantly expressed using Poisson brackets. This formulation not only simplifies calculations but also reveals deep connections between symmetries and conservation laws, as exemplified by .

Poisson Brackets and Their Properties

Poisson bracket and properties

Top images from around the web for Poisson bracket and properties

  • Liouville's theorem (Hamiltonian) - Wikipedia View original

    Is this image relevant?

  • Jacobian - Knowino View original

    Is this image relevant?

  • Liouville's theorem (Hamiltonian) - Wikipedia View original

    Is this image relevant?

1 of 3

Top images from around the web for Poisson bracket and properties

  • Liouville's theorem (Hamiltonian) - Wikipedia View original

    Is this image relevant?

  • Jacobian - Knowino View original

    Is this image relevant?

  • Liouville's theorem (Hamiltonian) - Wikipedia View original

    Is this image relevant?

1 of 3
  • definition mathematically expressed as f,g=∑i=1n(∂f∂qi∂g∂pi−∂f∂pi∂g∂qi){f,g} = \sum_{i=1}^n (\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}) quantifies change in one variable relative to another in phase space (position-momentum space)
  • Properties include antisymmetry f,g=−g,f{f,g} = -{g,f}, linearity af+bg,h=af,h+bg,h{af+bg,h} = a{f,h} + b{g,h}, fg,h=fg,h+gf,h{fg,h} = f{g,h} + g{f,h}, and f,g,h+g,h,f+h,f,g=0{f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0
  • qi,qj=0{q_i,q_j} = 0, pi,pj=0{p_i,p_j} = 0, and qi,pj=δij{q_i,p_j} = \delta_{ij} form basis for calculations

Canonical invariance of Poisson bracket

  • preserve form
  • Proof involves expressing new variables (Qi,Pi)(Q_i, P_i) in terms of old (qi,pi)(q_i, p_i), calculating Poisson brackets in new coordinates, and showing fundamental Poisson brackets remain unchanged
  • Invariance allows flexible coordinate choice in Hamiltonian mechanics (spherical, cylindrical)

Time Evolution and Conserved Quantities

Time evolution via Poisson bracket

  • Time evolution equation dfdt=f,H+∂f∂t\frac{df}{dt} = {f,H} + \frac{\partial f}{\partial t} with HH as system Hamiltonian
  • Applies to coordinates and momenta: qi˙=qi,H\dot{q_i} = {q_i,H} and pi˙=pi,H\dot{p_i} = {p_i,H}
  • Provides alternative derivation of Hamilton's equations
  • Facilitates easier calculation of complex variable time evolution ()
  • Offers coordinate-independent formulation enhancing problem-solving flexibility

Conserved quantities through Poisson bracket

  • Conserved quantities remain constant over time, satisfying f,H=0{f,H} = 0
  • Noether's theorem links symmetries to conserved quantities via Poisson brackets
  • Examples: angular momentum (central force), (time-independent systems)
  • Symmetries in Hamiltonian systems: translational (linear momentum), rotational (angular momentum), time-translation (energy)
  • Conservation check: calculate f,H{f,H} for quantity ff, if zero then ff conserved (total energy in isolated system)

Key Terms to Review (17)

Angular Momentum: Angular momentum is a physical quantity that represents the rotational momentum of a system. It is a vector quantity defined as the product of a particle's position vector and its linear momentum, mathematically expressed as $$ extbf{L} = extbf{r} imes extbf{p}$$, where $$ extbf{L}$$ is angular momentum, $$ extbf{r}$$ is the position vector, and $$ extbf{p}$$ is the linear momentum. This concept is vital for understanding both particle dynamics and rigid body motion, as it describes how the motion of objects changes when they rotate and interact with forces.
Antisymmetry: Antisymmetry refers to a property of certain mathematical operations or functions where swapping two elements results in a change of sign. In the context of Poisson brackets, antisymmetry plays a critical role in defining the relationship between observables in classical mechanics, ensuring that the brackets between two functions behave consistently under exchanges of their arguments. This property also supports the underlying structure of canonical invariants, reinforcing how physical quantities relate to one another under transformations.
Canonical transformations: Canonical transformations are a type of change of coordinates in phase space that preserve the form of Hamilton's equations. These transformations enable the transition between different sets of generalized coordinates and momenta while maintaining the underlying symplectic structure, which is vital for the consistency of Hamiltonian mechanics. Understanding these transformations is essential for solving complex systems and analyzing their properties through various methods like the Hamilton-Jacobi equation and Poisson brackets.
Conserved Quantities: Conserved quantities are physical properties of a system that remain constant over time, regardless of the internal or external forces acting upon it. These quantities are fundamental in both classical and quantum mechanics, playing a crucial role in understanding the behavior of systems through symmetries, governing equations, and the dynamics of periodic motions.
Energy: Energy is a physical quantity that represents the ability to perform work or produce change. In mechanics, it is a central concept, connecting different phenomena such as motion, force, and time, and it plays a crucial role in understanding the dynamics of systems through various forms like kinetic and potential energy.
Fundamental Poisson Brackets: Fundamental Poisson brackets are mathematical expressions that quantify the relationship between the dynamical variables of a system in Hamiltonian mechanics. They play a crucial role in understanding the structure of phase space and the evolution of physical systems, encapsulating how observables change over time. These brackets are essential for establishing canonical invariants, which preserve their form under transformations, thereby maintaining the symplectic structure of Hamiltonian systems.
Hamilton's equations: Hamilton's equations are a set of first-order differential equations that describe the evolution of a physical system in classical mechanics. They provide a reformulation of Newtonian mechanics, emphasizing energy conservation and enabling a clearer understanding of systems with multiple degrees of freedom. These equations are fundamental in connecting classical mechanics to other areas, such as field theory, adiabatic processes, and advanced mathematical techniques like Poisson brackets.
Hamiltonian: The Hamiltonian is an operator corresponding to the total energy of a system, encompassing both kinetic and potential energy. It plays a central role in classical and quantum mechanics by governing the evolution of a system over time. This operator is closely related to observables and their measurements, as it allows for the calculation of the dynamics of various physical systems, including phenomena such as spin-orbit coupling and the behavior of canonical coordinates in phase space.
Jacobi Identity: The Jacobi identity is a fundamental property of the Lie bracket in the context of Lie algebras and Poisson brackets, stating that for any three elements, the cyclic permutation of the bracket must equal zero. This property highlights the importance of symmetry and structure within mathematical systems, serving as a key condition for the algebraic structure of both classical mechanics and quantum mechanics.
Linearity: Linearity refers to the property of a mathematical system where the output is directly proportional to the input, allowing for superposition. This concept means that if you combine two inputs, the resulting output will be the sum of the outputs from each input applied individually. It is crucial in understanding various mathematical frameworks, such as vector spaces and transformations, and has important implications in both classical mechanics and quantum mechanics.
Noether's Theorem: Noether's Theorem is a fundamental principle in theoretical physics that connects symmetries and conservation laws, stating that every continuous symmetry of a physical system corresponds to a conserved quantity. This concept bridges various areas of mechanics and field theory, revealing how invariance under transformations leads to the conservation of energy, momentum, and other quantities.
Poisson bracket: The Poisson bracket is a mathematical operator used in classical mechanics that defines the relationship between two functions on phase space. It captures how one function changes as a result of the dynamics determined by another function, essentially describing how observables evolve over time. This concept is crucial for understanding the structure of Hamiltonian mechanics, especially in terms of canonical transformations and invariants.
Product Rule: The product rule is a fundamental principle in calculus that provides a method for finding the derivative of the product of two or more functions. This rule states that if you have two functions, say u(x) and v(x), the derivative of their product can be expressed as the derivative of the first function times the second function, plus the first function times the derivative of the second function. In the context of Poisson brackets and canonical invariants, the product rule helps in calculating derivatives involving products of position and momentum coordinates, leading to essential relationships in Hamiltonian mechanics.
Rotational symmetry: Rotational symmetry refers to a property of a shape or system where it looks the same after being rotated around a central point by a certain angle. This concept is crucial in understanding how physical systems behave, especially when analyzing their angular momentum, conservation laws, and invariance under transformations.
Time evolution: Time evolution refers to the change in the state of a dynamical system over time, often described mathematically using equations that capture how the system's variables progress as time passes. This concept is crucial for understanding both classical and quantum systems, as it provides a framework for predicting future behavior based on present conditions. In classical mechanics, time evolution can be expressed through Hamiltonian or Lagrangian formulations, while in quantum mechanics, it is handled through unitary operators acting on state vectors.
Time-translation symmetry: Time-translation symmetry refers to the invariance of physical laws under a shift in time, meaning that the fundamental behaviors of a system remain unchanged regardless of when an observation is made. This concept is crucial in both classical and quantum mechanics as it implies the conservation of energy, showcasing a deep connection between symmetries and conservation laws in physics.
Translational Symmetry: Translational symmetry refers to a property of a physical system where its laws remain invariant under spatial translations, meaning the system's behavior does not change when it is shifted in space. This concept is significant as it connects to various aspects of physics, including conservation laws and the underlying structure of physical theories, indicating that certain quantities are preserved when symmetry is present.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide