SICM: Chapter 3

by Carson Reynolds

Entitled Hamiltonian Mechanics, chapter 3 of SICM builds on the development provided in Chapter 1 of the Lagrangian. It begins by comparing the momenta-centric formulation of the Hamiltonian to the velocity-centric Lagrangian. Key are that Hamilton’s Equations are first-order formulations, while Lagrangians are second-order differential equations. The development of Hamilton’s equations is motivated by an effort to formulate it in terms of quantities that can be conserved, the moment and the energy.

Hamilton’s Equations are really just two equations giving the derivates of the state and the momenta. The Hamiltonian is defined as (H = PV-L). Chapter 3 provides three derivations. The first is shown by applying calculus and algebra to the Lagrangian Formulation. A second makes use of the Legendre Transformation. Given two functions F and G, the two are related by a Legendre transform if DF=(DG)^-1. Namely that the inverse of one is the derivative of the other. A third derivation is takesn directly from the action principle. Using the calculus of variations, we again find a path-distinguishing function, this time in terms of momenta.

The chapter also introduces Poisson Brackets as a way of simplifying the notation for Hamilton’s Equations. A Poisson bracket {F,H} is defined by:

{F,H} = dsub1F dsub2H – dsub2F dsub1H

Poisson brakets satisfy Jacobi’s identity.

The chapter goes on to show that solutions to time-independent systems with one degree of freedom can be found by quadratature (integration). Separatrix are defined as saddle points distinguishing regions of distinct behavior, like oscillations versus circulating.

Unlike the Lagrangian formulation, showing that mementum are conserved reduces the dimensionality of the phase space. In some cases the dimensionality of coupled ODEs is reduced by two, specifically when a coordinate does not appear in the Hamiltonian.

The chapter concedes that most problems do not have enough symmetries to be reduced to quadrature. Another interesting catch is that different phase-space descriptions can describe the same evolution.

Systems that can’t be reduced to quadrature can be studies using Poinacare’s surface of section technique. Using surface of sections we can study the structure of phase space: chaotic zones and islands of regular behavior.

Henon-Heiles provide a specific example of the application of classical mechanics to the study of astronomical behavior. Henon-Hiles discovered that chaotic trajectories are sensitive to small changes in initial conditions. Chaotic trajectories separate exponentially with time.

Liouville’s Theorem is also covered. It states that “phase fluid” is incompressible. Basically, volume is conserved as regions evolve using the map provided by dynamics. Poincare’s recurrence theorem builds off of Liouville’s Theorem to suggest that almost all (chaotic or regular) trajectories eventually return arbitrarily close to where they started. Hamiltonian systems do not have attractors.