Learning Objectives:

• Understand the need and motivation behind the Legendre transformation.
• Learn how Legendre transformation changes the dependent variables of a function.
• Apply the concept in classical mechanics to transition from Lagrangian to Hamiltonian formulation.

---

Key Concepts / Definitions:

• Legendre Transformation: A mathematical tool used to switch the dependent variable of a function from one quantity to its conjugate.
• Conjugate Variables: Pairs of variables like $(q, p)$ or $(x, y)$, where one is the derivative of the function with respect to the other.
• Hamiltonian Mechanics: A reformulation of classical mechanics that utilizes Legendre transformation to shift from velocity-based to momentum-based variables.

---

Theory and Explanation:

The Legendre transformation is a mathematical operation used when a function is defined in terms of a variable, but we want to re-express it in terms of its derivative instead. This is particularly useful in physics when the original variables are not the most convenient for analysis.

Motivation:

Imagine you have a function $f(x)$, but in many situations, you want to work with $y = \frac{df}{dx}$ instead of $x$. The Legendre transform lets you rewrite $f(x)$ in terms of $y$.

This switch is often helpful in thermodynamics (switching between internal energy and enthalpy), or in mechanics (transitioning from velocity to momentum).

Basic Idea:

Let $f(x)$ be a smooth, convex function. Define:

• $y = \frac{df}{dx}$
• Then the Legendre transform $g(y)$ is given by: \(g(y) = xy - f(x)\)

This $g(y)$ is a new function where $x$ is now expressed in terms of $y$. Importantly, $g(y)$ and $f(x)$ carry equivalent information but in different variables.

Application in Classical Mechanics:

In classical mechanics, we begin with the Lagrangian: \(L(q, \dot{q}, t)\) where:

• $q$ is the generalized coordinate,
• $\dot{q}$ is the generalized velocity,
• $t$ is time.

We define the generalized momentum as: \(p = \frac{\partial L}{\partial \dot{q}}\)

To switch from the Lagrangian (which depends on $\dot{q}$) to the Hamiltonian, we use the Legendre transformation: \(H(q, p, t) = p \dot{q} - L(q, \dot{q}, t)\)

Here, we re-express the dynamics in terms of $q$ and $p$, which are often more natural in physical systems. The resulting Hamiltonian describes the total energy of the system.

---

Solved Examples:

• Example 1 (Mathematical Function):
  Problem: Perform the Legendre transform of $f(x) = ax^2$.
  Solution:

  \[y = \frac{df}{dx} = 2ax \Rightarrow x = \frac{y}{2a}\] \[g(y) = xy - f(x) = \left(\frac{y}{2a}\right)y - a\left(\frac{y}{2a}\right)^2 = \frac{y^2}{2a} - \frac{y^2}{4a} = \frac{y^2}{4a}\]

• Example 2 (Mechanics: Lagrangian to Hamiltonian):
  Problem: For the Lagrangian $L = \frac{1}{2}m \dot{q}^2 - V(q)$, find the Hamiltonian.
  Solution:

  \[p = \frac{\partial L}{\partial \dot{q}} = m \dot{q} \Rightarrow \dot{q} = \frac{p}{m}\] \[H = p \dot{q} - L = \frac{p^2}{m} - \left( \frac{p^2}{2m} - V(q) \right) = \frac{p^2}{2m} + V(q)\]

• Example 3 (Thermodynamics: Internal Energy to Enthalpy):
  Problem: Derive enthalpy $H(S, P)$ from internal energy $U(S, V)$.
  Solution:

  Pressure is conjugate to volume: $P = -\left( \frac{\partial U}{\partial V} \right)_S$.
  Perform a Legendre transform:

  \[H(S, P) = U(S, V) + P V\]

• Example 4 (Thermodynamics: Internal Energy to Helmholtz Free Energy):
  Problem: Derive Helmholtz free energy $F(T, V)$ from internal energy $U(S, V)$.
  Solution:

  Temperature is conjugate to entropy: $T = \left( \frac{\partial U}{\partial S} \right)_V$
  Perform a Legendre transform:

  \[F(T, V) = U(S, V) - T S\]

• Example 5 (Thermodynamics: Internal Energy to Gibbs Free Energy):
  Problem: Derive Gibbs free energy $G(T, P)$ from internal energy $U(S, V)$.
  Solution:

  Perform two successive Legendre transforms:

  \[G(T, P) = U + P V - T S\]

• Example 6 (Thermodynamics: Helmholtz to Gibbs):
  Problem: Derive Gibbs free energy from Helmholtz free energy.
  Solution:

  \[G = F + P V = U - T S + P V\]

---

Important Points / Summary:

• The Legendre transformation replaces dependence on a variable with dependence on its conjugate.
• In mechanics, it allows a switch from velocity to momentum variables.
• The Hamiltonian formulation derived via Legendre transformation is essential in quantum mechanics and advanced classical physics.

---

Practice Questions:

• Short Answer:
  1. What is the Legendre transformation of $f(x) = e^x$?
  2. Define conjugate momentum and explain its role in Legendre transformation.
• Numerical:
  1. If $L = \frac{1}{2}m\dot{q}^2 + A\dot{q}$, find $H$.
  2. Find the Legendre transform of $f(x) = \ln x$.
• MCQs:
  1. The Legendre transformation is primarily used to:
     • (a) Integrate functions
     • (b) Switch variables from a function to its derivative
     • (c) Eliminate time from equations
     • (d) Solve differential equations
       Answer: (b)
  2. In classical mechanics, $p = \frac{\partial L}{\partial \dot{q}}$ is:
     • (a) Hamiltonian
     • (b) Energy
     • (c) Momentum
     • (d) Position
       Answer: (c)

---