I have a question about the discussion of constrained Lagrangian systems in the book Mathematical Aspects of Classical and Celestial Mechanics by Arnold et al. (section 1.2.5).

The Lagrangian system is defined on a manifold $M$. The constraints pick out a submanifold $S$ of the phase space $TM$, and are of the form $$f_1(q,\dot{q},t)=\cdots=f_n(q,\dot{q},t)=0,$$ with the requirement that $f_{1\dot{q}}'$, ..., $f_{n\dot{q}}'$ be linearly independent at each point. The virtual velocities of the state $(q,\dot{q})$ at time $t$ are the set of vectors $\xi\in T_qM$ satisfying $$f_{1\dot{q}}'\cdot \xi=\cdots=f_{n\dot{q}}'\cdot \xi=0.$$

I'm confused by these definitions. What if we want to impose a constraint $f$ that only depends on $q$? Then $f_{\dot{q}}'=0$, and all vectors at every point would be virtual velocities. But doesn't that mean there are no constraints on the system?

posted almost 4 years ago

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Technically Natural‭

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