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I was recently preparing for a test on Classical Mechanics and a friend of mine started wondering if there was any method through which we could obtain the restriction forces acting on a certain particle without using the Lagrange Multipliers method. At first glance my feeling would be that since Lagrangian Mechanics deals primarily with action and energies and not with forces, the answer would be no. But I'm really curious, any help would be appreciated.

Qmechanic
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Jorge Cabezut
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  • FWIW, if the constraints are holonomic one doesn't need Lagrange multipliers to solve the Lagrange equations. – Qmechanic Oct 25 '18 at 20:34
  • I see. Can restriction forces be calculated without issue in that case? I'm not trying to obtain equations of motion or conserved quantities. How would one go about obtaining the RF's in the case of holonomic restraints? – Jorge Cabezut Oct 26 '18 at 15:29

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  1. If the constraints are holonomic one doesn't need Lagrange multipliers to solve Lagrange equations $$ \frac{d}{dt}\frac{\partial T}{\partial \dot{q}^j}-\frac{\partial T}{\partial q^j}~=~Q^a_j,\qquad j~\in~\{1,\ldots, n\},\tag{1}$$ where $$Q^a_j~=~\sum_{i=1}^N {\bf F}^a_i\cdot \frac{\partial {\bf r}_i}{\partial q^j},\qquad j~\in~\{1,\ldots, n\},\tag{2}$$ is the $j$'th applied generalized force.

  2. The constraint force ${\bf F}^c_{i}$ on the $i$'th point particle with position $${\bf r}_i(q^1, \ldots, q^n,t),\qquad i~\in~\{1,\ldots, N\},\tag{3}$$ can then in principle be reconstructed via Newton's 2nd law $${\bf F}^c_{i}~=~\dot{\bf p}_i-{\bf F}^a_i,\tag{4}$$ where ${\bf F}_i^a$ is the applied force from eq. (2).

  3. See also this related Phys.SE post.

References:

  1. H. Goldstein, Classical Mechanics, Chapters 1 & 2.
Qmechanic
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