Legendre Transform

Legendre transform is a duality concept that comes up in building the dual theory of Hamiltonian mechanics from Lagrangian mechanics.  Legendre transform of a function on a space gives a function on the dual space. In mechanics, for instance, it gives the hamiltonian of the cotangent phase space from the lagrangian of the configuration space( tangent space).

It gives an alternative way to see real-valued functions for example. One can specify it as the value(height) of the function at a point or can specify the tangent line passing through the point on the graph. So we can choose to represent a function by the y-intercepts of the tangent lines.(We allow the function to take infinity as a value to include vertical lines). If f(x) was the function, we consider f*(p) as a value of negative of the y-intercept of the tangent line to y=f(x) with slope p. If there is at most one tangent of any slope we have a function f*(p). ( This happens for convex for example). This correspondence is called the Legendre transform.

For a general convex function we can describe this function f*(p) as sup_{x} p(x) -f(x). This definition is very general and once we have a space and function on the space, we can get the function on the dual function by similar definition.

Examples: f(x) = ax+b   …  f*(p)=-b if p=a else \infty.

f(x)=x^a/a f*(p)= p^b/b where 1/a+ 1/b =1.

f(x) =e^x, f*(p)=plogp-p

The two functions are related by the fact that their derivatives are inverses of each other. Infact this property describe the legendre transform under a certain normalisation. This description shows that these two functions are dual pairs i.e., the dual of the dual is the original function itself.(We need to identify the double dual of spaces with the space)

f–> f +c, f*—>f*-b,

f—> af, f*(p) —> af*(p/a)

f(x) —> f(x+a) , f*(p) —> f*(p)-ap

f(x)—>f(ax),  f*(p)—>f*(p/a)

f—>f^-1 ,.. f*(p) —> -pf*(1/p)

There are lots of similarities between the fourier transform which is related to the translation/ derivative operators and legendre transform. Just like the FT manily concerns the additive structure and has properties like the convolutions going to products , here in this case the Legendre transform mainly concern the convexity ( order structure) and properties like infimal convolution going to sums. This sum of how suggest the relevance of the tropical mathematics: The sum and product replaced by the minimum and sums.

In fact, it is true.  Legendre transform is the tropical analog of the Fourier transform. One important property of the LTs is that the local curvatures of the function and its dual are inversely related. (Hints at the uncertainty relations?) Why are there some similarities between the two transforms? What are the relationships between them? How does the semiring (R, min, +) relate to (R,+,x). All these questions are addressed in a field of mathematics called idempotent math.

Idempotent math is like a dequantization procedure. Quantizing objects by changing some kind of object another is a familiar thing in physics. The Lagrangian mechanics corresponds to Feynman path integrals. The hamiltonian mechanics to Heisenberg formulation of QM. The transformation from Lagrangian to Hamiltonian mechanics by Legendre transform corresponds to the Fourier transform of the wave function to get wave function in momentum space. And we have certain parameter h when deformed to zero along imaginary direction gives initial objects from the quantized objects. So idempotent math is to find such limit theories of math.

A dequantization procedure we describe is the Maslov dequantization which in the limit deforms the Ring  ( R,+,x) into the semiring (R+, min,+). For a parameter h, associate the additions and multiplications u sum_h v = h log(e^x/h + e^y/h), u \otimes_h  v = u+v.  As h tends to zero these operations give the semiring.