The principle of least action as a Lagrange variational problem: Stationarity and extremality conditions

Papastavridis, John G.

doi:10.1016/0020-7225(86)90072-8

Cited by 4 publications

(1 citation statement)

References 3 publications

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“…It is thus not surprising that trajectory stability is closely related to the character of the stationary trajectory action ͑saddle point or minimum͒. 23,[29][30][31] Planetary orbits also exhibit crossing points distant from the location of a disturbance; an incremental change in the velocity at one point in the orbit leads to initial and continued divergence of the two orbits which, for certain potential functions, reverses to bring them together again at a distant point. This later crossing point is defined as the kinetic focus for the Maupertuis action W applied to spatial orbits ͑see Appendix B͒.…”

Section: Why Worldlines Crossmentioning

confidence: 99%

When action is not least

Gray

Taylor

2007

American Journal of Physics

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We examine the nature of the stationary character of the Hamilton action S for a space-time trajectory ͑worldline͒ x͑t͒ of a single particle moving in one dimension with a general time-dependent potential energy function U͑x , t͒. We show that the action is a local minimum for sufficiently short worldlines for all potentials and for worldlines of any length in some potentials. For long enough worldlines in most time-independent potentials U͑x͒, the action is a saddle point, that is, a minimum with respect to some nearby alternative curves and a maximum with respect to others. The action is never a true maximum, that is, it is never greater along the actual worldline than along every nearby alternative curve. We illustrate these results for the harmonic oscillator, two different nonlinear oscillators, and a scattering system. We also briefly discuss two-dimensional examples, the Maupertuis action, and newer action principles.

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Section: Why Worldlines Crossmentioning

confidence: 99%

When action is not least

Gray

Taylor

2007

American Journal of Physics

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The variational principles of mechanics, and a reply to C. D. Bailey

Papastavridis

1987

Journal of Sound and Vibration

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The Least Action and the Metric of an Organized System

Georgiev

2002

Open Syst. Inf. Dyn.

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In this paper we formulate the Least Action Principle for an Organized System as the minimum of the total sum of the actions of all of the elements. This allows us to see how this most basic law of physics determines the development of the system towards states with less action -organized states. Also we state that the metric tensor can describe the specific state of the constraints of the system, which is its actual organization. With this the organization is defined in two ways: 1. A quantitative: the action I. 2. A qualitative: the metric tensor g µν. These two measures can describe the level of development and the specifics of the organization of a system. We consider closed and open systems.

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The principle of least action as a Lagrange variational problem: Stationarity and extremality conditions

Cited by 4 publications

References 3 publications

When action is not least

When action is not least

The variational principles of mechanics, and a reply to C. D. Bailey

The Least Action and the Metric of an Organized System

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