The Jacobi's principle of stationary full action and its consequences

The purpose of this article is to extend the applicability of the stationarity principle of the full Jacobi action to non-conservative natural systems and to derive equations of motion corresponding to this extended principle. To this end, in addition to the well-known variation of the Jacobi action with respect to coordinates, we propose independently variating time. Small variations in coordinates and time depend on the point on the true trajectory with the usual boundary conditions.

💡 Research Summary

The paper presents a comprehensive extension of Jacobi’s principle of stationary full action to non‑conservative natural systems. Starting from the standard Lagrangian for a natural system, (L=\frac12\mu_{\alpha\beta}\dot q^{\alpha}\dot q^{\beta}+P_{\alpha}\dot q^{\alpha}-U), the author derives the canonical momentum (p_{\alpha}=\mu_{\alpha\beta}\dot q^{\beta}+P_{\alpha}) and the total energy (E=\frac12\mu_{\alpha\beta}\dot q^{\alpha}\dot q^{\beta}+U). Using the relation (dt = dq/n) with (n=\sqrt{2T}) (where (T) is the kinetic energy), the time variable is eliminated in favour of the arc‑length parameter (q). This leads to a decomposition of the action into a “shortened Jacobi action” (S_J=\int(p,dq-E,dt)) and a residual term (S_R=-\int E,dt).

The novelty lies in allowing independent variations of both the generalized coordinates (q^{\alpha}) and the time function (t(q)). The variations are defined point‑wise along the true trajectory, with the usual fixed‑endpoint conditions (\delta q^{\alpha}(q_1)=\delta q^{\alpha}(q_2)=\delta t(q_1)=\delta t(q_2)=0). Under these conditions the total action (S=S_J+S_R) is stationary, i.e. (\delta S=0). This yields two fundamental relations:

1. Energy‑gradient relation
   (\partial T/\partial q^{\alpha} = -\partial U/\partial q^{\alpha}), which expresses the balance between kinetic‑energy gradients and potential‑energy gradients.

2. Geometric Jacobi relation
   (n,\partial n/\partial q^{\alpha} + \partial U/\partial q^{\alpha}=0), the familiar geometric form of Jacobi’s theorem.

Combining these, the author obtains a system of (2s+1) second‑order differential equations for the coordinates (q^{\alpha}) and the tangent vector (v^{\alpha}=dq^{\alpha}/ds) (with (s) the arc length). The system is fully determined by (2s) initial data: the initial kinetic energy (or total energy), the initial coordinates, and the initial tangent vector. Consequently, the trajectory in configuration space and its parametrisation by arc length are uniquely fixed.

From the derived equations the standard Lagrange equations are reconstructed, now containing explicit non‑conservative contributions: \