# Hamilton-Ostrogradski principle

*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