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## Homework Statement

I need some help understanding a derivation in a textbook. It involves the Lagrangian in generalized coordinates.

## Homework Equations

The text states that generalized coordinates {q_1, ..., q_3N} are related to original Cartesian coordinates [tex]q_\alpha = f_\alpha(\mathbf r_1, ..., \mathbf r_N)[/tex]

Makes sense to me...

Velocities in general coordinates are, by the chain rule,

[tex]\dot{\mathbf r_i} = \sum_{\alpha=1}^{3N} \frac{\partial \mathbf r_i}{\partial q_\alpha} \dot{q_\alpha}[/tex]

Ok so then kinetic energy in cartesian is

[tex] K = \frac{1}{2} \sum_{i=1}^N m_i \dot{r_i}^2 [/tex]

So KE in new velocities is then

[tex] \tilde{K}(q, \dot{q}) = \frac{1}{2} \sum_{\alpha=1}^{3N} \sum_{\beta=1}^{3N}[\sum_{i=1}^{N} m_i \frac{\partial \mathbf r_i}{\partial q_\alpha}\cdot \frac{\partial \mathbf r_i}{\partial q_\beta}] \dot{q_\alpha} \dot{q_\beta} [/tex]

[tex] = \frac{1}{2} \sum_{\alpha=1}^{3N} \sum_{\beta=1}^{3N} G_{\alpha\beta}(q_1, ..., q_3N) \dot{q_\alpha} \dot{q_\beta}[/tex]

where the text calls the expression in brackets the mass metric tensor.

Ok, so then given this new KE, the Lagrangian (KE - PE) is

[tex]L = \frac{1}{2} \sum_{\alpha=1}^{3N} \sum_{\beta=1}^{3N} G_{\alpha}{\beta}(q_1,...,q_{3N})\dot{q_\alpha} \dot{q_\beta} - U(\mathbf r_1(q_1,...,q_{3N}), ..., \mathbf r_N(q_1, ... , q_{3N})) [/tex]

All of the above makes sense to me, but then the text substitutes the lagrangian into the Euler-lagrange equation and gets the equation of motion for each [itex]q_\gamma, \gamma = 1, ... 3N[/itex]

[tex]\sum_{\beta=1}^{3N} G_{\gamma\beta}(q_1,...,q_{3N})\ddot{q_\beta} + \sum_{\alpha=1}^{3N} \sum_{\beta=1}^{3N} [\frac{\partial G_{\gamma\beta}}{\partial q_\alpha} - \frac{\partial G_{\alpha\beta}}{\partial q_\gamma} ]\dot{q_\alpha}\dot{q_\beta} = -\frac{\partial U}{\partial q_\gamma} [/tex]

(Euler-lagrange: [itex]\frac{d}{dt}(\frac{\partial L}{\partial \dot{q_\gamma}}) - \frac{\partial L}{\partial q_\gamma} = 0[/itex])

## The Attempt at a Solution

Ok, so I can see where the first and third term in the equation of motion are coming from. The first is a result of [itex]\frac{d}{dt}(\frac{\partial L}{\partial \dot{q_\gamma}})[/itex] acting on the kinetic energy part of the Lagrangian. The third term is the result of [itex]\frac{\partial L}{\partial q_\gamma}[/itex] acting on the potential energy part of the Lagrangian. The mass metric tensor is a function of the q's, so I feel like the second term is coming from [itex]\frac{\partial L}{\partial q_\gamma}[/itex] acting on the kinetic energy term, but that's where I'm stuck.

Where does [itex]\frac{\partial G_{\gamma\alpha}}{\partial q_\alpha}[/itex] come from?

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