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Generalized forces

The work done on the dynamical system when its Cartesian coordinates change by $ \delta x_j$ is simply

$\displaystyle \delta W = \sum_{j=1,{\cal F}} f_j\,\delta x_j$ (7.3)

Here, the $ f_j$ are the Cartesian components of the forces acting on the various particles making up the system. Thus, $ f_1, f_2, f_3$ are the components of the force acting on the first particle, $ f_4, f_5, f_6$ the components of the force acting on the second particle, and so on. Using Equation (7.2), we can also write

$\displaystyle \delta W = \sum_{j=1,{\cal F}} f_j\sum_{i=1,{\cal F}}\frac{\partial x_j}{\partial q_i}\,\delta q_i.$ (7.4)

The preceding expression can be rearranged to give

$\displaystyle \delta W = \sum_{i=1,{\cal F}} Q_i\,\delta q_i,$ (7.5)

where

$\displaystyle Q_i = \sum_{j=1,{\cal F}} f_j\,\frac{\partial x_j}{\partial q_i}.$ (7.6)

Here, the $ Q_i$ are termed generalized forces. A generalized force does not necessarily have the dimensions of force. However, the product $ Q_i\,q_i$ must have the dimensions of work. Thus, if a particular $ q_i$ is a Cartesian coordinate then the associated $ Q_i$ is a force. Conversely, if a particular $ q_i$ is an angle then the associated $ Q_i$ is a torque.

Suppose that the dynamical system in question is conservative. It follows that

$\displaystyle f_j = -\frac{\partial U}{\partial x_j},$ (7.7)

for $ j=1,{\cal F}$ , where $ U(x_1,x_2,\cdots,x_{\cal F},t)$ is the system's potential energy. Hence, according to Equation (7.6),

$\displaystyle Q_i = - \sum_{j=1,{\cal F}} \frac{\partial U}{\partial x_j}\,\frac{\partial x_j}{\partial q_i} = - \frac{\partial U}{\partial q_i},$ (7.8)

for $ i=1,{\cal F}$ .


next up previous
Next: Lagrange's equation Up: Lagrangian mechanics Previous: Generalized coordinates
Richard Fitzpatrick 2016-03-31