14.1 Hamiltonian systems and symplectic structures

Hamiltonian systems and symplectic structures are powerful tools for studying classical mechanics. They use generalized coordinates and momenta to describe a system's evolution through phase space, offering insights into conservation laws and system behavior.

This approach provides a geometric framework for understanding dynamical systems. By exploring concepts like canonical transformations, Poisson brackets, and Liouville's theorem, we gain a deeper understanding of the underlying mathematical structure of classical mechanics.

Hamiltonian Mechanics and Phase Space

Formulation and Characteristics of Hamiltonian Mechanics

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• Hamiltonian mechanics reformulates classical mechanics using generalized coordinates and momenta ($q_i$ and $p_i$) instead of Cartesian coordinates and velocities
• Describes the evolution of a system through a function called the Hamiltonian ($H(q,p,t)$), which represents the total energy of the system
• Hamiltonian equations of motion:
  • $\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}$
  • $\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}$
• Advantages of Hamiltonian mechanics include symmetry, conservation laws, and ease of handling constraints

Phase Space and Its Properties

• Phase space is a 2n-dimensional space where each point represents a unique state of a system with n degrees of freedom
  • Generalized coordinates ($q_i$) and momenta ($p_i$) form the axes of phase space
• Trajectories in phase space represent the evolution of a system over time (harmonic oscillator)
• Volume in phase space is conserved due to Liouville's theorem, which states that the phase space density remains constant along the system's trajectories (incompressible flow)

Canonical Transformations and Poisson Brackets

• Canonical transformations are coordinate transformations that preserve the form of Hamilton's equations
  • Examples include point transformations, extended point transformations, and completely canonical transformations
• Generating functions are used to find canonical transformations and express the relationship between old and new variables (F1, F2, F3, F4)
• Poisson brackets are a mathematical operation that measures the change in a function due to the evolution of the system
  • Defined as $\{f,g\} = \sum_{i=1}^n \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$
• Properties of Poisson brackets include antisymmetry, linearity, and the Jacobi identity

Symplectic Structures and Conserved Quantities

Symplectic Manifolds and Their Properties

• A symplectic manifold is a smooth manifold equipped with a closed, non-degenerate 2-form called the symplectic form ($\omega$)
  • In canonical coordinates, the symplectic form is written as $\omega = \sum_{i=1}^n dq_i \wedge dp_i$
• Symplectic manifolds are the natural setting for Hamiltonian mechanics, as they provide a geometric framework for describing the evolution of a system
• Properties of symplectic manifolds include:
  • Even dimensionality (2n)
  • Existence of a symplectic form
  • Darboux's theorem, which states that locally, all symplectic manifolds are equivalent to the standard symplectic structure on $\mathbb{R}^{2n}$

Liouville's Theorem and Its Implications

• Liouville's theorem states that the phase space volume is conserved under the flow of a Hamiltonian system
  • Mathematically, $\frac{d}{dt} \int_{\Omega(t)} d^nq d^np = 0$, where $\Omega(t)$ is a region in phase space evolving with time
• Implications of Liouville's theorem include:
  • Incompressibility of phase space flow
  • Preservation of phase space density
  • Connection to the concept of entropy in statistical mechanics (microcanonical ensemble)

Integrable Systems and Action-Angle Variables

• An integrable system is a Hamiltonian system with n degrees of freedom that possesses n independent conserved quantities (first integrals) in involution
  • Two functions f and g are said to be in involution if their Poisson bracket vanishes: $\{f,g\} = 0$
• Integrable systems exhibit regular, non-chaotic motion and can be solved analytically (Kepler problem, harmonic oscillator)
• Action-angle variables are a special set of canonical coordinates for integrable systems
  • Action variables (J) are conserved quantities related to the phase space area enclosed by the system's trajectories
  • Angle variables ($\theta$) evolve linearly with time and describe the position of the system along its trajectory
• In action-angle variables, the Hamiltonian depends only on the action variables, simplifying the equations of motion:
  • $\frac{dJ_i}{dt} = -\frac{\partial H}{\partial \theta_i} = 0$
  • $\frac{d\theta_i}{dt} = \frac{\partial H}{\partial J_i} = \omega_i(J)$