Suppose I have a mechanical system with l + m degrees of freedom and an associated lagrangian(adsbygoogle = window.adsbygoogle || []).push({});

[itex]L(\alpha,\beta,\dot{\alpha},\dot{\beta},t)[/itex]

where [itex]\alpha\in\mathbb{R}^l[/itex] and [itex]\beta\in\mathbb{R}^m[/itex].

Now suppose I have a known [itex]\mathbb{R}^l[/itex]-valued function f(t) and define a new lagrangian

[itex]M(\beta,\dot{\beta},t)=L(f(t),\beta,\dot{f}(t), \dot{\beta},t)[/itex]

Do the equations that derive from M correctly describe the motion of the initial mechanical system, where the first l degrees of freedom are constrained to the motion f(t) (by means of an external force)?

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# Time dependent lagrangian

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