Hamilton's Equations of Motion on Phase Space

A hierarchy of translational moment equations for the position and momentum observables is considered. The resulting set of equations is truncated in a manner that includes only first and second order moments. This closure procedure is consistent with the conservation of energy and, for spherically symmetric potentials, with the conservation of angular momentum. The method is valid for both classical and quantal mechanics with the only difference being that for quantal systems, the dispersions in momentum and position must satisfy Heisenberg's uncertainty principle. With finite dispersion in the position, the average position and momentum do not satisfy Hamilton's equations of motion since the average acceleration is modulated by the variance–covariance in the position. For classical systems, the second order moments may be taken to zero in which case Hamilton's equations are obtained. For quantal systems, a comparision is made with Heller's Gaussian wave packet approach. The moment method is illustrated by applying it to the reflection of a phase space packet from a one dimensional repulsive inverse square potential. Applications to the calculation of collision cross sections are envisaged.

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The Hamiltonian Formulation of Classical Mechanics: Hamilton's Equations of Motion

Fundamental Principles of Classical Mechanics ◽

10.1142/9789814551496_0033 ◽

2014 ◽

pp. 381-397

Keyword(s):

Equations Of Motion ◽

Hamiltonian Formulation ◽

Classical Mechanics ◽

Hamilton's Equations ◽

Hamilton’S Equations

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Hamilton’s equations of motion of a vortex filament in the rotating Bose–Einstein condensate and their “soliton” solutions

JETP Letters ◽

10.1134/s0021364016120109 ◽

2016 ◽

Vol 103 (12) ◽

pp. 780-784 ◽

Cited By ~ 6

Author(s):

V. P. Ruban

Keyword(s):

Equations Of Motion ◽

Soliton Solutions ◽

Einstein Condensate ◽

Vortex Filament ◽

Bose Einstein Condensate ◽

Bose Einstein ◽

Hamilton's Equations ◽

Hamilton’S Equations

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Observable and Unobservable Mechanical Motion

Entropy ◽

10.3390/e22070737 ◽

2020 ◽

Vol 22 (7) ◽

pp. 737

Author(s):

J. Gerhard Müller

Keyword(s):

Information Gain ◽

Mechanical Energy ◽

Equations Of Motion ◽

Minimum Energy ◽

Radiation Energy ◽

Mechanical Motion ◽

Radiation Emission ◽

Physical Action ◽

Hamilton's Equations ◽

Hamilton’S Equations

A thermodynamic approach to mechanical motion is presented, and it is shown that dissipation of energy is the key process through which mechanical motion becomes observable. By studying charged particles moving in conservative central force fields, it is shown that the process of radiation emission can be treated as a frictional process that withdraws mechanical energy from the moving particles and that dissipates the radiation energy in the environment. When the dissipation occurs inside natural (eye) or technical photon detectors, detection events are produced which form observational images of the underlying mechanical motion. As the individual events, in which radiation is emitted and detected, represent pieces of physical action that add onto the physical action associated with the mechanical motion itself, observation appears as a physical overhead that is burdened onto the mechanical motion. We show that such overheads are minimized by particles following Hamilton’s equations of motion. In this way, trajectories with minimum curvature are selected and dissipative processes connected with their observation are minimized. The minimum action principles which lie at the heart of Hamilton’s equations of motion thereby appear as principles of minimum energy dissipation and/or minimum information gain. Whereas these principles dominate the motion of single macroscopic particles, these principles become challenged in microscopic and intensely interacting multi-particle systems such as molecules moving inside macroscopic volumes of gas.

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Vibrational/Rotational Energy Levels

Unimolecular Reaction Dynamics ◽

10.1093/oso/9780195074949.003.0004 ◽

1996 ◽

Author(s):

Tomas Baer ◽

William L. Hase

Keyword(s):

Rotational Motion ◽

Energy Levels ◽

Equations Of Motion ◽

Rotational Energy ◽

Unimolecular Reaction ◽

Coordinate Systems ◽

Electronic Motion ◽

Hamilton's Equations ◽

Hamilton’S Equations

The first step in a unimolecular reaction is the excitation of the reactant molecule’s energy levels. Thus, a complete description of the unimolecular reaction requires an understanding of such levels. In this chapter molecular vibrational/rotational levels are considered. The chapter begins with a discussion of the Born-Oppenheimer principle (Eyring, Walter, and Kimball, 1944), which separates electronic motion from vibrational/ rotational motion. This is followed by a discussion of classical molecular Hamiltonians, Hamilton’s equations of motion, and coordinate systems. Hamiltonians for vibrational, rotational, and vibrational/rotational motion are then discussed. The chapter ends with analyses of energy levels for vibrational/rotational motion. The Born-Oppenheimer principle assumes separation of nuclear and electronic motions in a molecule. The justification in this approximation is that motion of the light electrons is much faster than that of the heavier nuclei, so that electronic and nuclear motions are separable.

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Hamilton’s Equations & Routhian Reduction

10.1093/oso/9780198822370.003.0016 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Angular Momentum ◽

Phase Space ◽

Poisson Bracket ◽

First Integrals ◽

Late Nineteenth Century ◽

Algebra Structure ◽

French Mathematician ◽

Late Nineteenth ◽

Hamilton's Equations ◽

Hamilton’S Equations

In this chapter, the Poisson bracket and angular momentum are investigated and first integrals are used to develop conservation laws as a canonical Noether’s theorem. The Poisson bracket was developed by the French mathematician Poisson in the late nineteenth century and it is a reformulation, or at least a tidying up, of Hamilton’s equations into one neat package. The Poisson bracket of a quantity with the Hamiltonian describes the time evolution of that quantity as one moves along a curve in phase space. The Lie algebra structure of symmetries in mechanics is highlighted using this formulation. The classical propagator is derived using the Poisson bracket.

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