Why use the Legendre transform over back substitution

[nashed]

In thermodynamics we use a variation of the Legandere Legendre transform to move from one description of the system to another ( depending on what is the control variable...), but I don't understand why choose to use the Legandere Legendre transform over writing x in terms of s=dy/dx and back substituting to y=f(x), please refrain from using differential forms if you can, as much as I tried I just don't seem to have the foundations to understand them (being a physics student) and unfortunately I really don't have the time to try and cover everything needed to understand them.

Mod note: Fixed misspelling of Legendre.

 

[jvicens]

Thank you for your question about the use of the Legandere Legendre transform in thermodynamics. I understand that you are a physics student and may not have a strong foundation in differential forms. I will try my best to explain the reasoning behind using this transformation without using differential forms.

The Legandere Legendre transform is a mathematical tool that allows us to switch between different descriptions of a system by changing the control variable. In thermodynamics, we often encounter situations where we need to change the control variable to better understand the behavior of the system. This is where the Legandere Legendre transform becomes useful.

For example, let's say we have a system described by a variable x and we want to change the control variable to y. We can do this by using the Legandere Legendre transform, which involves taking the derivative of the function with respect to x and then substituting it into the original function. This results in a new function in terms of y, which gives us a different description of the system.

Now, you may be wondering why we choose to use the Legandere Legendre transform instead of simply writing x in terms of y. The reason for this is that in thermodynamics, we often encounter situations where the control variable is not a physical quantity, but rather a mathematical construct. In these cases, it is not possible to write x in terms of y directly.

Additionally, the Legandere Legendre transform allows us to easily switch between different descriptions of the system without having to go through the process of solving for x in terms of y. This can be very useful in complex systems where the equations may be difficult to solve.

I hope this explanation helps you understand the importance of the Legandere Legendre transform in thermodynamics. While it may seem complicated at first, it is a powerful tool that allows us to better understand and analyze systems. If you have any further questions, please do not hesitate to ask.
Scientist in Thermodynamics