14.1 Hamiltonian systems and symplectic structures
3 min read•august 7, 2024
Hamiltonian systems and symplectic structures are powerful tools for studying . They use generalized coordinates and momenta to describe a system's evolution through , 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 , 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 (qi and pi) 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:
dtdqi=∂pi∂H
dtdpi=−∂qi∂H
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 (qi) and momenta (pi) 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}=∑i=1n(∂qi∂f∂pi∂g−∂pi∂f∂qi∂g)
Properties of Poisson brackets include antisymmetry, linearity, and the Jacobi identity
Symplectic Structures and Conserved Quantities
Symplectic Manifolds and Their Properties
A is a smooth manifold equipped with a closed, non-degenerate 2-form called the (ω)
In canonical coordinates, the symplectic form is written as ω=∑i=1ndqi∧dpi
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
, which states that locally, all symplectic manifolds are equivalent to the standard symplectic structure on R2n
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, dtd∫Ω(t)dnqdnp=0, where Ω(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 vanishes: {f,g}=0
exhibit regular, non-chaotic motion and can be solved analytically (Kepler problem, harmonic oscillator)
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 (θ) 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: