# Jacobi principle

principle of stationary action

An integral variational principle in mechanics that was established by C.G.J. Jacobi  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 ) 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.

How to Cite This Entry:
Jacobi principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_principle&oldid=32721
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article