Some examples and consequences of the properties considered are demonstrated and discussed using the simple model of two. A legendre transformation is a way of transforming a function of some variable into another function of another variable without losing any information. Legendre transformation and thermodynamics physics stack. The notation for the partial legendre transform, indicating replacement of both the systems entropy, s, and volume, v, by their conjugate variables, the temperature, t, and minus pressure. The relationship is given by the legendre transformation. In classical mechanics, it was used to find the hamiltonian from lagrangian. Abstract legendre transforms appear in two places in a standard undergraduate physics curriculum. The aim of this report is to list and explain the basic properties of the legendre fenchel transform, which is a generalization of the legendre transform commonly encountered in physics. Legendre transforms in chemical thermodynamics sciencedirect. Examples of the legendre transform relationship in physics. Y p of y x, which corresponds to replacing the extensive parameter x by its conjugate intensive parameter p. Generating function for legendre polynomials if a is a. In thermodynamics, it is usual to employ a slight variant, di ering by a sign from the usual legendre transform.
In mathematics, legendre transform is an integral transform named after the mathematician adrienmarie legendre, which uses legendre polynomials as kernels of the transform. The lagrangian function l is a function of coordinates and velocities, but it is often more convenient to define the hamiltonian function h with a legendre transform. In thermodynamics this elimination is assured by the criterion of stability expressing a dependence of p on x. To change to another set of natural variables, a new function is defined by subtracting off a particular conjugate pair. The generalized definition acts on the larger space of nonconvex function cylindrical volume, but still maps it down to the space of convex ones. The legendre transformation is discussed in detail in part iv, where it plays an important role in plasticity theory, and other topics. Xwith thermodynamic identity dydxdxintroducing the variable xconjugate to paired with x. S, v where is the energy, s is entropy and v is volume. Pdf use of legendre transforms in chemical thermodynamics. Legendre transform is a special case of jacobi transform. It is commonly used in classical mechanics to derive the hamiltonian formalism out of the lagrangian formalism and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of. The legendre transform green arrow is an involution on the space of convex functions blue.
The legendre transformation of l is its fiber derivative. Legendre transforms in chemical thermodynamics semantic scholar. Manogue, david roundy, and elizabeth gire 1department of physics, oregon state university, corvallis, or 97331 one commonly and persistently di cult topic in thermodynamics is the relationship between. Legendre transforms in chemical thermodynamics 2223 series in complex mixtures of hydrocarbons under various conditions, and i recognized that i could set the partial pressure of ethylene at some specified value and recalculate the standard gibbs energies of. Because of the restrictions we have placed on the original function we are transforming and therefore on all its legendre transforms, we know that mixed partial derivatives will be equal. Legendre transforms 100203 in class and during recitation, we have seen how legendre transforms allow us to change the independent variables that we need to control in any experimental condition that we may encounter in practice. A legendre transform has made a major contribution in mechanics. Legendre transformation explained with animation youtube. Me346a introduction to statistical mechanics wei cai. The function is shown in red, and the tangent line at.
For example, the thermodynamic potentials defined by legendre transforms 1. Appendixa thermodynamic functions and legendre transforms background chapter 2 prepared a brief thermodynamic basis for understanding transitions from. For example, the equation of state of ideal gas can be written as. These functions appear as solutions of boundary value problems in physics and engineering. Legendre transformation an overview sciencedirect topics.
Jan 10, 20 trying an example from a textbook but i dont understand legendre transform at all. This function is referred to as the legendre transform y p. The legendre fenchel transform generalizes the legendre transform for the nondifferentiable or nonconvex functions, by finding a onetoone correspondence between the independent variable x and the derivative of fx. In thermodynamics the legendre transformation brings about a transition from some state functions to others, for example from the specific volume. In a legendre transform, to replace one independent variable with its conjugate, a new function. Department of physics and astronomy, university of maine, orono, maine 04469. It is commonly used in classical mechanics to derive the hamiltonian formalism out of the lagrangian formalism and in thermodynamics to derive the thermodynamic potentials, as.
Diagram illustrating the legendre transformation of the function fx. Yet, in typical undergraduate or graduate courses, the power of motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by fourier transforms. The second and third relations imply the rst one up to a constant, and the rst one xes the constant. The legendre transformation can be extended to arbitrary element typesusing a corresponding inner product. Heat is nothing but a special form of energy mayer 1842 first law \conservation of energy statement 1. We include the example here only as a stepping stone to the legendre transform in statistical mechanics and thermodynamics. Legendres transformation is obtained by reordering the respective terms. The definition of legendre transformation is a little complicate. Legendre transformations many important thermodynamic equations are closely related. The legendre transform in modern optimization roman a. The natural variables for u are s and v, which means that if the function us,v is known for a given system we can obtain all thermodynamic properties of the system through the di. The legendre transformation can be generalized to the legendrefenchel transformation.
You have to remember that the statements that helmholtz free energy is minimized at equilbrium and gibbs free energy is minimized at equilbrium are referring to two different notions of minimizing. The legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. While its often used, it may not be always so wellbehaved. The legendre transformation connects two ways of specifying the same physics, via functions of two related \conjugate variables. Ghorai 1 lecture xiii legendre equation, legendre polynomial 1 legendre equation this equation arises in many problems in physics, specially in boundary value problems. So, according to the definition, this is the legendre transform of g induced by the transformation y x, y.
Legendre transform in twodimensional thermodynamics haynes miller fall, 2008 the legendre transform is a useful device for producing new state functions from old ones. We wish to perform a legendre transformation on es,v to change the s variable to something else its going to be the. Wellknown and important concepts such as the legendre polynomials and legendre transformation are named after him. The understanding of the difference between the two transforms is important be9 cause the general transform which arises in statistical mechanics is the legendrefenchel 10 transform, not the legendre transform. A legendre transform is a procedure for expressing the information content of some function by using a different independent. In thermodynamics, the internal energy u can be legendre transformed into various. It is commonly used in thermodynamics and in the hamiltonian formulation of classical mechanics. A new interpretation of legendres transformation and. The use of legendre transforms is not restricted to thermodynamics. In practical applications, for example, it is impossible to have the entropy of the system at. In mathematics and physics, the legendre transformation, named after adrienmarie legendre, is an involutive transformation on the realvalued convex functions of one real variable.
Euler 1776 if is a finitedimensional function that is smooth, strictly convex, and increases at infinity faster than a linear function, the legendre transform can be defined thus. Juan carlos marrero et al, a survey of lagrangian mechanics and control on lie algebroids and lie groupoids. Legendre transformation of a singlevariable function. The legendrefenchel transform generalizes the legendre transform for the nondifferentiable or nonconvex functions, by finding a onetoone correspondence between the independent variable x and the derivative of fx. In typical undergraduate and graduate courses the motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by fourier transforms.
Use of legendre transforms in chemical thermodynamics 57 t able 1 conjugate pairs of variables in work terms for the fundamental equation for u a. The procedure of transformation corresponds to integration by parts. It leaves invariant the small set of functions red whose first derivatives are their own inverse, like and. Legendre transformation is the transformation of a convex function, which produces a new function of a new variable. One can perform a legendre transform to change the variable from x to x2 using dx3 3. Trying an example from a textbook but i dont understand legendre transform at all.
In this abstract setting, the legendre transformation corresponds to the tautological oneform. The legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics, statistical mechanics, and thermodynamics. Legendre transform is a linear change in variables in which one or more products of conjugate variables are subtracted from the internal energy to define a new thermodynamic potential 216. As is to be expected, if the hamiltonian was generated by a legendre transformation from a lagrangian, then this lagrangian can be recovered. Since the legendre transformation is selfinverse, it is interesting to study what will happen if it is applied to the hamiltonian. Legendre transformations in basic mechanics and thermodynamics, expressed in the standard way. Legendre transform the legendre transformation is used extensivelyin mechanics taking us from lagrangians to hamiltonians and back and thermodynamics relating energy to helmholtz. The legendre transformation can be generalized to the legendre fenchel transformation. Jun 28, 2017 the above equation holds for each pair satisfying, so we can explicitly confirm that 3 it is also clear from looking at a graph that a function inverse is symmetric i. Polyak abstract the legendre transform let is a product of a general duality principle.
We will deal with partial derivatives and legendre transforms. Discussion of legendre transformation in the context of lie algebroids is in. Ordinarily, the inverse of a transformation is distinct from the transform itself. Kennerly, a graphical derivation of the legendre transform, pdf online at. Both terms are in common use, and are denoted by gt,p,ni. This is latex version of short text tutorial on legendre transform from sci. R n is a chart on tm, and for any point x, v in this chart, the fiber derivative and hence also the legendre transformation of l is defined by. Legendre transformation explained with animation duration. Legendre transforms and their application in thermodynamics.
Since we can consider each pair of variables separately i. From \\delta s 0\ from postulate 2, to the euler formula, legendre transforms, and finally maxwell relations and jacobians, you have all the problemsolving tools thermodynamics has to offer to tackle equilibrium problems. Table 1 shows some examples of legendre transformations in basic mechanics and thermodynamics, expressed in the standard way. The legendre transform distinguishes itself in that it is its own inverse. Check that f satis es the existenceuniqueness conditions. In the 2nd lecture, we will discuss the mathematics of thermodynamics, i. Though this is still a correct differentiation relation. With reference to thermodynamics, callen 3 discusses application of legendre transformations. Natural variables are important because when a thermodynamic potential can be determined as a function of its natural variables, all the other thermodynamic properties of the system can be obtained by taking partial derivatives. 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. In thermodynamics, a thermodynamic system is completely described by knowing its fundamental equation in energetic form. Physical meaning of legendre transformation stack exchange.
In reality, the different ways to describe the system that differ by a legendre transformation are equally fundamental or equally natural so its often useful to be familiar with all of them and to know what is the relationship between them. It is informative to understand such legendre transformations in a more general context. However, discussions of it tend to be ad hoc, poorly motivated, and confusing. For example, an inverse laplace transform is not given by the same formula. The rst law of thermodynamics establishes the total energy as a state variable. This equality of mixed partials leads to all the maxwell relations. Legendre transforms appear in two places in a standard undergraduate physics curriculum. R is a lower semicontinuous proper convex function, then its legendre transform f. Sep 26, 2016 homework help service the legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics, statistical mechanics. Legendre transform in twodimensional thermodynamics. A graphical derivation of the legendre transform sam kennerly april 12, 2011 this work is licensed under the creative commons attribution 3.
The tangent line intersects the vertical axis at 0. In this chapter we summarize information about several functions which are widely used for mathematical modeling in engineering. Us,v,n hs, p, n, from the internal energy us,v,n to. Legendre transform from the lagrangian l to the hamiltonian h. Fx for example, the inner product of complex matrices is. The legendre transform is a transformation from a convex differentiable function fx to a function that depends on the family of tangents s. As an example of this, consider the helmholtz free energy, ft. Legendre transformation wikimili, the free encyclopedia.
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