Jacobi principle

principle of stationary action

An integral variational principle in mechanics that was established by C.G.J. Jacobi [1] for holonomic conservative systems. According to the Jacobi principle, if the initial position $P_0$ and the final position $P_1$ of a holonomic conservative system are given, then for the actual motion the Jacobi action

$$S=\int\limits_{P_0}^{P_1}\sqrt{2(U+h)}ds,\quad ds^2=\sum_{i,j=1}^na_{ij}dq_idq_j$$

has a stationary value in comparison with all other infinitely-near motions between $P_0$ and $P_1$ with the same constant value $h$ of the energy as in the actual motion. Here $U(q_1,\ldots,q_n)$ is the force function of the active forces on the system, and $q_i$ are the generalized Lagrange coordinates of the system, whose kinetic energy is

$$T=\frac12\sum_{i,j=1}^na_{ij}\dot q_i\dot q_j,\quad\dot q_i\equiv\frac{dq_i}{dt}.$$

Jacobi proved (see [1]) that if $P_0$ and $P_1$ are sufficiently near to one another, then for the actual motion the action $S$ has a minimum. The Jacobi principle reduces the problem of determining the actual trajectory of a holonomic conservative system to the geometrical problem of finding, in a Riemannian space with the metric

$$ds^2=\sum_{i,j=1}^na_{ij}dq_idq_j,$$

an extremal of the variational problem.

See also Variational principles of classical mechanics.

References

[1]	C.G.J. Jacobi, "Vorlesungen über Dynamik" , G. Reimer (1884)

References

[a1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)