Jacobi's Formulation
For many systems, the kinetic energy is quadratic in the generalized velocities
although the mass tensor may be a complicated function of the generalized coordinates . For such systems, a simple relation relates the kinetic energy, the generalized momenta and the generalized velocities
provided that the potential energy does not involve the generalized velocities. By defining a normalized distance or metric in the space of generalized coordinates
one may immediately recognize the mass tensor as a metric tensor. The kinetic energy may be written in a massless form
or, equivalently,
Hence, the abbreviated action can be written
since the kinetic energy equals the (constant) total energy minus the potential energy . In particular, if the potential energy is a constant, then Jacobi's principle reduces to minimizing the path length in the space of the generalized coordinates, which is equivalent to Hertz's principle of least curvature.
Read more about this topic: Maupertuis' Principle
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