Velocities as function of canonical momenta

[jostpuur]

Assuming I've understood some claims correctly, having defined the canonical momenta with equation

$$p_k = \frac{\partial L}{\partial \dot{q}_k},$$

we can solve the velocities as functions

$$\dot{q}_k(q_1,\ldots,q_n,p_1,\ldots, p_n)$$

precisely when the determinant

$$\textrm{det}\Big(\Big(\frac{\partial^2 L}{\partial \dot{q}_k\partial\dot{q}_{k'}}\Big)_{k,k'\in\{1,\ldots,n\}}\Big)$$

is non-zero. Why is this the case? The result looks reasonable, but I have difficulty seeing where this is coming from.

 

[CompuChip]

Basically, to solve the velocities, you will want to use the implicit function theorem, from which it follows that the Hessian must be invertible. Also see the exact statement in the link.

 

[charng]

The determinant in question is known as the Hessian matrix, and it represents the second-order derivatives of the Lagrangian with respect to the generalized velocities. This matrix is crucial in determining the behavior of a system described by the Lagrangian, as it contains information about the curvature and extrema of the Lagrangian.

In the context of classical mechanics, the equations of motion can be obtained from the principle of least action, which states that the true path of a system is one that minimizes the action integral. This integral is given by the difference between the initial and final values of the Lagrangian, integrated over time. By varying the path of the system and setting the resulting variation of the action to zero, we can obtain the equations of motion.

In order for this variation to be well-defined, the Hessian matrix must be invertible, or in other words, its determinant must be non-zero. This ensures that the system has a unique solution and that the equations of motion can be derived from the principle of least action.

Furthermore, the Hessian matrix also plays a key role in determining the stability of a system. A positive-definite Hessian matrix indicates a stable system, while a negative-definite matrix indicates an unstable system. This is important in understanding the behavior of a physical system and predicting its future evolution.

In summary, the non-zero determinant of the Hessian matrix is crucial in determining the behavior and dynamics of a system described by the Lagrangian. It ensures the uniqueness of the solution and provides information about the stability of the system.