Jacobi Action Principle

A Second Variational Principle for Conservative Systems

Hamilton.'s action principle in classical mechanics is widely taught. There is a lesser known, but important Jacobi principle which is like Fermat's principle for waves. This form of action principle was used by Schrodinger to arrive at his hafous equation for qunatum mechanics of a point particles.

The Euler Lagrange action principle says that the equations of motion of a particle can be formulated as a requiremnet that the action be minimum. The classical particle trajectory is given by the path which minimizes the action

\begin{equation}

S = \int_{t_1}^{t_2} L ( q, \dot{q})\, dt

\end{equation}

with end points fixed. Here \(L = T-V\) is the Lagrangian,\(T\) kinetic energy and \(V\) being the potential energy.Let us assume that the kinetic energy is given by

\begin{equation}

T = \sum_{i,k=1}^n a_{ik} \dot{q}_i \dot{q}_k .

\end{equation}

For a conservative system, the variational principle can also be formulated as trajectories as being curves of shortest length. where the distance

between two points is given by the metric

\begin{equation}

ds^2 =(E-V)\sum_{i,k=1}^n a_{ik} dq_i dq_k

\end{equation}

where \(E\) is the energy of the particle. This variational principle is known as Jacobi action principle and is closer to the Fermat principle for light.

References

- F. Gnatmacher,
*Lectures in Analytical Dynamics*, Mir Publisher Moscow (1970) - Cornelius Lanczos,
*The Variational Principles of Mechanics*, 4th edn, Dover Publications Inc., New York (1986) - L. D. Landau and E. M. Lifshitz,
*Mechanics*, 3rd edn, Elsevier, New Delhi, {1976}