Relativistic non-Hamiltonian mechanics

Journal Article · Fri Oct 15 04:00:00 EDT 2010 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2010.06.011· OSTI ID:21452953

Relativistic particle subjected to a general four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u{sub {mu}u}{sup {mu}} + c{sup 2} = 0, where c is the speed of light in vacuum. In the general case, four-forces are non-potential, and the relativistic particle is a non-Hamiltonian system in four-dimensional pseudo-Euclidean space-time. We consider non-Hamiltonian and dissipative systems in relativistic mechanics. Covariant forms of the principle of stationary action and the Hamilton's principle for relativistic mechanics of non-Hamiltonian systems are discussed. The equivalence of these principles is considered for relativistic particles subjected to potential and non-potential forces. We note that the equations of motion which follow from the Hamilton's principle are not equivalent to the equations which follow from the variational principle of stationary action. The Hamilton's principle and the principle of stationary action are not compatible in the case of systems with nonholonomic constraint and the potential forces. The principle of stationary action for relativistic particle subjected to non-potential forces can be used if the Helmholtz conditions are satisfied. The Hamilton's principle and the principle of stationary action are equivalent only for a special class of relativistic non-Hamiltonian systems.

• OSTI ID: 21452953
• Journal Information: Annals of Physics (New York), Journal Name: Annals of Physics (New York) Journal Issue: 10 Vol. 325; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
ELECTROMAGNETIC RADIATION
ENERGY RANGE
EQUATIONS
EQUATIONS OF MOTION
EUCLIDEAN SPACE
FOUR-DIMENSIONAL CALCULATIONS
HAMILTONIANS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
POTENTIALS
QUANTUM MECHANICS
QUANTUM OPERATORS
RADIATIONS
RELATIVISTIC RANGE
RIEMANN SPACE
SPACE
SPACE-TIME
VARIATIONAL METHODS
VELOCITY
VISIBLE RADIATION