6.2 Poisson brackets and canonical invariants

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 antisymmetry and linearity, and form the basis for understanding time evolution and conserved quantities in physical systems.

Time evolution in Hamiltonian 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 Noether's theorem.

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?

• 

  dynamical systems - Proof involving Poisson bracket - Mathematics Stack Exchange 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?

• 

  dynamical systems - Proof involving Poisson bracket - Mathematics Stack Exchange 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?

• 

  dynamical systems - Proof involving Poisson bracket - Mathematics Stack Exchange 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?

• 

  dynamical systems - Proof involving Poisson bracket - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

• Poisson bracket definition mathematically expressed as ${f,g} = \sum_{i=1}^n (\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i})$ quantifies change in one variable relative to another in phase space (position-momentum space)
• Properties include antisymmetry ${f,g} = -{g,f}$, linearity ${af+bg,h} = a{f,h} + b{g,h}$, product rule ${fg,h} = f{g,h} + g{f,h}$, and Jacobi identity ${f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0$
• Fundamental Poisson brackets ${q_i,q_j} = 0$, ${p_i,p_j} = 0$, and ${q_i,p_j} = \delta_{ij}$ form basis for calculations

Canonical invariance of Poisson bracket

• Canonical transformations preserve Hamilton's equations form
• Proof involves expressing new variables $(Q_i, P_i)$ in terms of old $(q_i, p_i)$, 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 $\frac{df}{dt} = {f,H} + \frac{\partial f}{\partial t}$ with $H$ as system Hamiltonian
• Applies to coordinates and momenta: $\dot{q_i} = {q_i,H}$ and $\dot{p_i} = {p_i,H}$
• Provides alternative derivation of Hamilton's equations
• Facilitates easier calculation of complex variable time evolution (angular momentum)
• Offers coordinate-independent formulation enhancing problem-solving flexibility

Conserved quantities through Poisson bracket

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