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There is vast literature about constraint dynamics, but the available material is quite abstract and it feels to read about things which are far from reality.

Is there a simple example, where I can study a particular case, where

$\frac{\partial^2L}{\partial \dot q_i \partial \dot q_j}$

is singular and the velocities $\dot q_i$ cannot be expressed by the canonical momenta $p_i$?

I would like to see an example, where a primary constraint gives rise to secondary constraints. But since the Lagrangian must obviously be linear in velocities, I cannot figure out what that could be.

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  • $\begingroup$ This post (v1) seems like a list question since there is not a unique answer. $\endgroup$ – Qmechanic Jun 11 '17 at 7:04
  • $\begingroup$ I removed what you regarded as list. $\endgroup$ – michael Jun 11 '17 at 7:13
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The most famous example of what you want is the symmetrized Dirac Lagrangian:

enter image description here

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Actually I found good examples here.

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