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Difference between revisions of "Hamilton-Ostrogradski principle"

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''principle of stationary action''
 
''principle of stationary action''
  
 
A general integral variational principle of classical mechanics (cf. [[Variational principles of classical mechanics|Variational principles of classical mechanics]]), established by W. Hamilton [[#References|[1]]] for holonomic systems restricted by ideal stationary constraints, and generalized by M.V. Ostrogradski [[#References|[2]]] to non-stationary geometrical constraints. According to this principle, in a real motion of the system acted upon by potential forces,
 
A general integral variational principle of classical mechanics (cf. [[Variational principles of classical mechanics|Variational principles of classical mechanics]]), established by W. Hamilton [[#References|[1]]] for holonomic systems restricted by ideal stationary constraints, and generalized by M.V. Ostrogradski [[#References|[2]]] to non-stationary geometrical constraints. According to this principle, in a real motion of the system acted upon by potential forces,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046250/h0462501.png" /></td> </tr></table>
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$$S=\int\limits_{t_0}^{t_1}(T-U)dt=\int\limits_{t_0}^{t_1}Ldt$$
  
has a stationary value as compared with near, kinetically-possible, motions, with initial and final positions of the system and times of motion identical with those for the real motion. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046250/h0462502.png" /> is the kinetic energy, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046250/h0462503.png" /> is the potential energy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046250/h0462504.png" /> is the Lagrange function of the system. In certain cases the true motion corresponds not only to a stationary point of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046250/h0462505.png" />, but corresponds to its smallest value. For this reason the Hamilton–Ostrogradski principle is sometimes called the principle of least action. In the case of non-potential active forces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046250/h0462506.png" /> the condition of stationary action, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046250/h0462507.png" />, is replaced by the condition
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has a stationary value as compared with near, kinetically-possible, motions, with initial and final positions of the system and times of motion identical with those for the real motion. Here, $T$ is the kinetic energy, $U$ is the potential energy and $L=T-U$ is the Lagrange function of the system. In certain cases the true motion corresponds not only to a stationary point of the functional $S$, but corresponds to its smallest value. For this reason the Hamilton–Ostrogradski principle is sometimes called the principle of least action. In the case of non-potential active forces $F_v$ the condition of stationary action, $\delta S=0$, is replaced by the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046250/h0462508.png" /></td> </tr></table>
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$$\int\limits_{t_0}^{t_1}\left(\delta T+\sum_vF_v\cdot\delta r_v\right)dt=0.$$
  
 
====References====
 
====References====

Latest revision as of 14:39, 21 August 2014

principle of stationary action

A general integral variational principle of classical mechanics (cf. Variational principles of classical mechanics), established by W. Hamilton [1] for holonomic systems restricted by ideal stationary constraints, and generalized by M.V. Ostrogradski [2] to non-stationary geometrical constraints. According to this principle, in a real motion of the system acted upon by potential forces,

$$S=\int\limits_{t_0}^{t_1}(T-U)dt=\int\limits_{t_0}^{t_1}Ldt$$

has a stationary value as compared with near, kinetically-possible, motions, with initial and final positions of the system and times of motion identical with those for the real motion. Here, $T$ is the kinetic energy, $U$ is the potential energy and $L=T-U$ is the Lagrange function of the system. In certain cases the true motion corresponds not only to a stationary point of the functional $S$, but corresponds to its smallest value. For this reason the Hamilton–Ostrogradski principle is sometimes called the principle of least action. In the case of non-potential active forces $F_v$ the condition of stationary action, $\delta S=0$, is replaced by the condition

$$\int\limits_{t_0}^{t_1}\left(\delta T+\sum_vF_v\cdot\delta r_v\right)dt=0.$$

References

[1] W. Hamilton, , Report of the 4-th meeting of the British Association for the Advancement of Science , London (1835) pp. 513–518
[2] M. Ostrogradski, Mem. Acad. Sci. St. Petersbourg , 8 : 3 (1850) pp. 33–48


Comments

In English-language literature this principle goes by the name of Hamilton principle.

References

[a1] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
How to Cite This Entry:
Hamilton-Ostrogradski principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hamilton-Ostrogradski_principle&oldid=33044
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article