Exploring Roger Penrose's Functional Derivative Notation

[Rasalhague]

Roger Penrose, in The Road to Reality, introduces the idea of what he calls a "functional derivative", "denoted by using [itex]\delta[/itex] in place of [itex]\partial[/itex]; "Carrying out a functional derivative in practice is essentially just applying the same rules as for ordinary calculus" (Vintage 2005, p. 487). He cites expressions of the form

[tex]\frac{\delta \mathcal{L}}{\delta \Psi}[/tex]

presumably the partial derivative of the Lagrangian with respect to a function [itex]\Psi[/itex], specifically a tensor or spinor field on spacetime.

On p. 489, he uses the expression [itex]\delta S[/itex] to indicate that a quantity S has zero derivative with respect to all independent variables ("constituent fields").

On pp. 460-461, he describes [itex]\delta M[/itex] as the mass within an infinitesimal volume [itex]\delta V[/itex]. Is there any relationship between this notation and his later functional derivative delta? Is this, in some sense, an instance of a functional derivative, or just a coincidence of notation? When I first saw it, I guessed it might mean an integrand in general, and that he was using [itex]\delta V[/itex], say, for a volume element, rather than the traditional [itex]d V[/itex], so as to avoid the misleading impression that this was the exterior derivative of some covariant alternating tensor, V.

 

[jvicens]

I find Penrose's use of the functional derivative notation to be quite interesting and useful in understanding the concept of functional derivatives. The notation \delta is commonly used in mathematics to represent a small variation or change in a quantity. In the context of functional derivatives, it represents a small variation in a functional, which is a function of functions.

In the case of \frac{\delta \mathcal{L}}{\delta \Psi}, the functional derivative represents the change in the Lagrangian with respect to the function \Psi. This notation is useful in the field of physics, where the Lagrangian is used to describe the dynamics of a system. By taking the functional derivative, we can find the equations of motion for the system.

On the other hand, the notation \delta S, as explained by Penrose, represents a quantity with zero derivative with respect to all independent variables. This can be seen as a special case of a functional derivative, where the functional is the action of the system.

As for the use of \delta M to represent the mass within an infinitesimal volume, I believe it is just a coincidence of notation. This notation is commonly used in the field of physics to represent a small change in mass, and it is not related to functional derivatives.

In conclusion, Penrose's use of the functional derivative notation is a powerful tool in understanding the concept of functional derivatives in physics. It allows us to express the change in a functional with respect to its constituent functions, and it is a useful notation in many areas of physics. However, it is important to note that not all uses of the symbol \delta in physics are related to functional derivatives.