Definition of generalized force in Lagrangian formalism

Question

In some texts (e.g. Taylor's Classical Mechanics), the generalized force is defined to be (I'll simplify to one particle in one dimension for ease of notation): $Q \equiv \frac{\partial{L}}{\partial{q}}$.

While other texts (e.g. Goldstein's Classical Mechanics) define the generalized force to be: $Q \equiv F\frac{\partial{x}}{\partial{q}}$.

These two definitions are equivalent if the kinetic energy is independent of generalized coordinates $q$, but this is not the case in general. Adopting the former definition makes sense because then Lagrange's equation can be written as: $\dot{p} = Q$, which maintains the form of Newton's Second Law, where $p$ is the generalized momentum. On the other hand the latter definition makes sense because then it is the case that: $dW = Q dq$, which maintains the form of the definition of work in the generalized coordinates.

Are one of these two definitions ``correct''? Nowhere can I find this distinction discussed in my available texts; if one adopts the latter definition (generalized force = $F\frac{\partial{x}}{\partial{q}}$), then what is $\frac{\partial{L}}{\partial{q}}$ called?

Answer 1

TL;DR: The settings/frameworks and the definitions of generalized force in Refs. 1 and 2 are different.

1. Ref. 1 defines the generalized force as $$Q_j~:=~\frac{\partial L}{\partial q^j}\tag{7.15}$$ such that Euler-Lagrange (EL) equation looks like Newton's 2nd law: $\frac{dp_j}{dt}=Q_j$. This mostly$^1$ seems useful in the context of a stationary action principle (SAP), but possibly beyond point mechanics.

2. Ref. 2 defines the generalized force as $$Q_j~:=~\sum_{i=1}^N{\bf F}_i\cdot \frac{\partial {\bf r}_i}{\partial q^j}.\tag{1.49}$$ We can then derive Lagrange equations from d'Alembert's principle. This makes sense in the context of point mechanics even without a SAP and even with non-conservative and semi-holonomic constraints.

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$^1$ In applications beyond a SAP, definition (1.49) appears naturally, so that it seems confusing to also introduce definition (7.15). For the record we note that Ref. 1 assumes a SAP.

References:

1. J.R. Taylor, Classical Mechanics, 2005; eq. (7.15).

2. H. Goldstein, Classical Mechanics; eq. (1.49).