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## Summary:

- Stuck in Goldstein's Classical Mechanics

## Main Question or Discussion Point

I'm working my way through Goldstein's Classical Mechanics and have followed the arguments until section 2.4 (Extending Hamilton's Principle to Systems with Constraints). In the second paragraph, Goldstein states that "When we derive Lagrange's equations from either Hamilton's or D'Alembert's principle, the holonomic constraints appear in the last step when the variations in the ##q_i## were considered independent of eachother." I believe this refers back to equation 2.17: $$\delta J = \int_1^2\sum_i\bigg(\frac{\partial f}{\partial y_i} - \frac{d}{dx}\frac{\partial f}{\partial \dot{y}_i}\bigg)\delta y_i dx$$ with ##\delta y_i = \big(\frac{\partial y_i}{\partial \alpha}\big)_0 d\alpha##. I understand why deriving the Lagrange equations requires the variations to be independent. However, Goldstein then says "the virtual displacements in the ##\delta q_I##'s may not be consistent with the constraints. If there are ##n## variables and ##m## constraint equations ##f_\alpha## of the form ##f(\mathbf{r}_1, \mathbf{r}_2,...,t) = 0##, the extra virtual displacements are eliminated by the method of

I have more questions about the derivation that follows, but first of all, I don't understand what it means for the virtual displacements in the ##\delta q_I##'s to not be consistent with the constraints. Maybe if I clear this up, the remaining argument will be clear. Thanks!

*Lagrange undetermined multipliers*. "I have more questions about the derivation that follows, but first of all, I don't understand what it means for the virtual displacements in the ##\delta q_I##'s to not be consistent with the constraints. Maybe if I clear this up, the remaining argument will be clear. Thanks!

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