Can someone explains me the Legrendre transformation?

In summary, the Legrendre transformation is a mathematical operation used in classical mechanics and thermodynamics to derive the Hamiltonian from the Lagrangian and obtain other thermodynamic potentials such as the Gibbs free energy or Enthalpy. It is performed by applying the Legendre transform to a function of several variables. However, the specific process of computing with the Legendre transform may vary depending on the context. 3. The Legendre transform also has a geometrical meaning, relating to tangent spaces and manifolds, but this may not be relevant for understanding its application in thermodynamics. For more information, Wikipedia is a good reference source.
  • #1
fluidistic
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I first met the Legrendre transformation in classical mechanics (self study), I did not pay much attention of how the Hamiltonian is derived from the Lagrangian, by applying a Legrendre transform to the Lagrangian. Now I meet this transformation again in Thermodynamics when one has the internal energy and wants to obtain for example the Gibbs free energy or Enthalpy, etc.
Now I try to understand exactly how this transform (is it an operator?) acts on a function of several variables. I must of course be mistaken but I see in wikipedia what happens in the case of a single variable, not several variables.
Thus, if you give me a function of several variables, I have no idea how to take the Legendre transform(s) of this function.
Could someone explain me more or less what is it, how to apply it or at least give me a good reference (wikipedia would do the job if I knew where to look at exactly)?
Thanks!
 
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  • #2
There are a couple answers you might be looking for, so let me know which one you're after:
(1) How to compute with the Legendre Transform (like you might have to in Thermo).
(2) What is the meaning (usually) of the Legendre Transform
(3) The geometrical meaning of the Legendre Transform (thinking of tangent spaces, manifolds etc).

Cheers,

Kevin
 
  • #3
homology said:
There are a couple answers you might be looking for, so let me know which one you're after:
(1) How to compute with the Legendre Transform (like you might have to in Thermo).
(2) What is the meaning (usually) of the Legendre Transform
(3) The geometrical meaning of the Legendre Transform (thinking of tangent spaces, manifolds etc).

Cheers,

Kevin

Thanks a lot for your help!
1 and 2.
 

1. What is the purpose of the Legrendre transformation?

The Legrendre transformation is used to transform a function from one set of independent variables to another set of independent variables. This transformation is useful in simplifying equations and making them easier to solve.

2. How is the Legrendre transformation mathematically defined?

The Legrendre transformation is defined as the transformation of a function f(x) to a new function g(y), where y is a new independent variable, through the substitution of x with y and f'(x) with g'(y), such that g'(y) = x.

3. Can you provide an example of the Legrendre transformation?

One example of the Legrendre transformation is transforming the function f(x) = x^2 to g(y) = y^2 - 2y. This is done by substituting x with y and f'(x) with g'(y) = 2y - 2.

4. What are the benefits of using the Legrendre transformation?

The Legrendre transformation can simplify complicated equations and make them easier to solve by transforming them into a new set of variables. It can also help to identify symmetries and patterns in the equations.

5. In what fields of science is the Legrendre transformation commonly used?

The Legrendre transformation is commonly used in physics, especially in thermodynamics, and in mathematics to solve optimization problems. It is also used in other fields such as engineering and economics.

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