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I do not see how (2.34) shows that forces of semi-holonomic constraints do no work in the displacements ##\delta q_k## between the varied path and the actual path.

Starting from (2.31), I seem to be able to prove that such forces do do work, in contrary to what is claimed in the paragraph following (2.34).

Using ##Q^{(c)}## to denote the generalised force of constraint, (2.31) becomes

##\frac{d}{dt}\frac{\partial L}{\partial\dot{q_k}}-\frac{\partial L}{\partial q_k}=Q_k^{(c)}##

What follows is basically the reverse of the derivation of Lagrange's equation from D'Alembert's principle, as shown from (1.44) to (1.57).

Using ##L=T-V## and assuming ##V=V(q, t)##,

##\frac{d}{dt}\frac{\partial T}{\partial\dot{q_k}}-\frac{\partial T}{\partial q_k}+\frac{\partial V}{\partial q_k}-Q_k^{(c)}=0##

##\frac{d}{dt}\frac{\partial T}{\partial\dot{q_k}}-\frac{\partial T}{\partial q_k}-Q_k^{(a)}-Q_k^{(c)}=0##, where ##Q^{(a)}## is the generalised applied force.

##\Sigma_k\Big[\frac{d}{dt}\frac{\partial T}{\partial\dot{q_k}}-\frac{\partial T}{\partial q_k}-Q_k^{(a)}-Q_k^{(c)}\Big]\delta q_k=0##

##\Sigma_k\Big[\frac{d}{dt}[\frac{\partial}{\partial\dot{q_k}}(\Sigma_i\frac{1}{2}m_iv_i^2)]-\frac{\partial}{\partial q_k}(\Sigma_i\frac{1}{2}m_iv_i^2)-Q_k^{(a)}-Q_k^{(c)}\Big]\delta q_k=0##

##\Sigma_k\Big[\Sigma_i[\frac{d}{dt}(m_iv_i\cdot\frac{\partial v_i}{\partial\dot{q_k}})-m_iv_i\cdot\frac{\partial v_i}{\partial q_k}]-Q_k^{(a)}-Q_k^{(c)}\Big]\delta q_k=0##

##\Sigma_k\Big[\Sigma_i[\frac{d}{dt}(m_i\dot{r_i}\cdot\frac{\partial r_i}{\partial q_k})-m_i\dot{r_i}\cdot\frac{d}{dt}\frac{\partial r_i}{\partial q_k}]-Q_k^{(a)}-Q_k^{(c)}\Big]\delta q_k=0##

##\Sigma_k\Sigma_i\Big[(m_i\ddot{r_i}\cdot\frac{\partial r_i}{\partial q_k}-(F_i^{(a)}+F_i^{(c)})\cdot\frac{\partial r_i}{\partial q_k}\Big]\delta q_k=0##

##\Sigma_i\Big[m_i\ddot{r_i}\cdot\delta r_i-(F_i^{(a)}+F_i^{(c)})\cdot\delta r_i\Big]=0##

##\Sigma_i(F_i^{(a)}+F_i^{(c)}-\dot{p_i})\cdot\delta r_i=0##

If the virtual work done by ##F_i^{(c)}## is ##0##, we have

##\Sigma_i(F_i^{(a)}-\dot{p_i})\cdot\delta r_i=0##, which is (1.45)

Using the same steps as the derivation from (1.45) to (1.57), (that is, the reverse of what was done earlier) we get (1.57), which is different from (2.31).

We thus conclude that forces of semi-holonomic constraints must do work in virtual displacements.

What's wrong?

Below is the derivation of Lagrange's equation using D'Alembert's principle:

Starting from (2.31), I seem to be able to prove that such forces do do work, in contrary to what is claimed in the paragraph following (2.34).

Using ##Q^{(c)}## to denote the generalised force of constraint, (2.31) becomes

##\frac{d}{dt}\frac{\partial L}{\partial\dot{q_k}}-\frac{\partial L}{\partial q_k}=Q_k^{(c)}##

What follows is basically the reverse of the derivation of Lagrange's equation from D'Alembert's principle, as shown from (1.44) to (1.57).

Using ##L=T-V## and assuming ##V=V(q, t)##,

##\frac{d}{dt}\frac{\partial T}{\partial\dot{q_k}}-\frac{\partial T}{\partial q_k}+\frac{\partial V}{\partial q_k}-Q_k^{(c)}=0##

##\frac{d}{dt}\frac{\partial T}{\partial\dot{q_k}}-\frac{\partial T}{\partial q_k}-Q_k^{(a)}-Q_k^{(c)}=0##, where ##Q^{(a)}## is the generalised applied force.

##\Sigma_k\Big[\frac{d}{dt}\frac{\partial T}{\partial\dot{q_k}}-\frac{\partial T}{\partial q_k}-Q_k^{(a)}-Q_k^{(c)}\Big]\delta q_k=0##

##\Sigma_k\Big[\frac{d}{dt}[\frac{\partial}{\partial\dot{q_k}}(\Sigma_i\frac{1}{2}m_iv_i^2)]-\frac{\partial}{\partial q_k}(\Sigma_i\frac{1}{2}m_iv_i^2)-Q_k^{(a)}-Q_k^{(c)}\Big]\delta q_k=0##

##\Sigma_k\Big[\Sigma_i[\frac{d}{dt}(m_iv_i\cdot\frac{\partial v_i}{\partial\dot{q_k}})-m_iv_i\cdot\frac{\partial v_i}{\partial q_k}]-Q_k^{(a)}-Q_k^{(c)}\Big]\delta q_k=0##

##\Sigma_k\Big[\Sigma_i[\frac{d}{dt}(m_i\dot{r_i}\cdot\frac{\partial r_i}{\partial q_k})-m_i\dot{r_i}\cdot\frac{d}{dt}\frac{\partial r_i}{\partial q_k}]-Q_k^{(a)}-Q_k^{(c)}\Big]\delta q_k=0##

##\Sigma_k\Sigma_i\Big[(m_i\ddot{r_i}\cdot\frac{\partial r_i}{\partial q_k}-(F_i^{(a)}+F_i^{(c)})\cdot\frac{\partial r_i}{\partial q_k}\Big]\delta q_k=0##

##\Sigma_i\Big[m_i\ddot{r_i}\cdot\delta r_i-(F_i^{(a)}+F_i^{(c)})\cdot\delta r_i\Big]=0##

##\Sigma_i(F_i^{(a)}+F_i^{(c)}-\dot{p_i})\cdot\delta r_i=0##

If the virtual work done by ##F_i^{(c)}## is ##0##, we have

##\Sigma_i(F_i^{(a)}-\dot{p_i})\cdot\delta r_i=0##, which is (1.45)

Using the same steps as the derivation from (1.45) to (1.57), (that is, the reverse of what was done earlier) we get (1.57), which is different from (2.31).

We thus conclude that forces of semi-holonomic constraints must do work in virtual displacements.

What's wrong?

Below is the derivation of Lagrange's equation using D'Alembert's principle:

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