What is Legendre transformation in thermodynamics?

A Legendre transform converts from a function of one set of variables to another function of a conjugate set of variables. Both functions will have the same units. All of these thermodynamic potentials have units of energy.

What is Legendre transformation in physics?

The Legendre transformation connects two ways of specifying the same physics, via functions of two related (“conjugate”) variables. mechanics, the Lagrangian L and Hamiltonian H are Legendre transforms of each other, depending on conjugate variables ˙x (velocity) and p (momentum) respectively.

What is importance of Legendre transformation?

The Legendre transform shows how to define a function that contains the same infor- mation as F x but as a function of dF/dx. is a strictly monotonic function of x because this character- ization also permits us to treat functions whose negative is convex.

How do you spell Legendre?

A·dri·en Ma·rie [a-dree-an ma-ree], 1752–1833, French mathematician.

What is Lagrange equation of motion?

One of the best known is called Lagrange’s equations. The Lagrangian L is defined as L = T − V, where T is the kinetic energy and V the potential energy of the system in question.

How do you write the Lagrange equation?

What is the meaning of Legendre?

French: status name from Old French gendre ‘son-in-law’, with the definite article le. See also Gendron.

What did Adrien Marie Legendre do?

Adrien-Marie Legendre, (born September 18, 1752, Paris, France—died January 10, 1833, Paris), French mathematician whose distinguished work on elliptic integrals provided basic analytic tools for mathematical physics.

How do you find the Legendre transform of a function?

The Legendre transform is linked to integration by parts, pdx = d(px) − xdp . Let f be a function of two independent variables x and y, with the differential Assume that it is convex in x for all y, so that one may perform the Legendre transform in x, with p the variable conjugate to x.

Is the Legendre transform of a convex function invertible?

The Legendre transform of a convex function is convex. with a non zero (and hence positive, due to convexity) double derivative. . Then . Thus, f ′ ( x ) = p . {\\displaystyle f^ {\\prime } (x)=p~.} is itself differentiable with a positive derivative and hence strictly monotonic and invertible. .

Is the Legendre transform nothing but the origin ordinate?

So, intuitively we see that the Legendre transform is nothing but the origin ordinate of the slope of fat x. It is obvious-at least graphically- that we can recover fknowing ψ⁢(m). We now prove it rigourously. Theorem 1(Invertibility and duality of Legendre Transformation).

In this abstract setting, the Legendre transformation corresponds to the tautological one-form . The strategy behind the use of Legendre transforms in thermodynamics is to shift from a function that depends on a variable to a new (conjugate) function that depends on a new variable, the conjugate of the original one.