Generalized force


Generalized force

The idea of a Generalized Force is a concept stemming from Lagrangian mechanics. It is a consequence of the application of generalized coordinates to a system undergoing acceleration.

When a particle undergoes a virtual displacement delta mathbf{r} under the influence of a force mathbf{F} the virtual work done by that force is given by:


delta W = mathbf{F} cdot delta mathbf{r} = sum_{i} F_i delta x_i.

Translating to generalized coordinates:


delta W = sum_{i} (sum_{j=1}^n F_i frac {partial x_i}{partial q_j} delta q_j),

and by reversing the order of summation we get


delta W = sum_{j=1}^n ( sum_{i}F_i frac {partial x_i}{partial q_j})delta q_j.

It is from this formulation that the idea of a generalized force stems. The above equation can be written as


delta W = sum_{j=1}^n (Q_j)delta q_j

where


Q_j = sum_{i}F_i frac {partial x_i}{partial q_j}

is called the generalized force associated with the coordinate q_j.

Since Q_jq_j has the dimension of work, Q_j will have the dimension of force if q_j is a distance, and the dimension of torque if q_j is an angle.

ee also

*Lagrangian mechanics
*Generalized coordinates
*Degrees of freedom (physics and chemistry)
*Virtual work


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