Legendre Transform Calculator

Introduction

The Legendre transform is a fundamental tool in mathematics and physics that converts a convex function into its dual representation. It is widely used in fields such as classical mechanics, thermodynamics, and optimization. This calculator allows you to compute the Legendre transform of a single-variable convex function at a specified slope value.

Understanding the Legendre Transform

Given a real-valued convex function $f\left(x\right)$, the Legendre transform $f^*\left(p\right)$ is defined as:

This means we look for the value of $x$ that maximizes the expression $px-f(x)$. Intuitively, this corresponds to finding the point where the line with slope $p$ is tangent to the graph of $f$. The Legendre transform then records the intercept of this tangent line.

Mathematical Characterization

For differentiable convex functions, the maximizing $x$ satisfies the condition:

Once this $x$ is found, the Legendre transform value is computed as:

How to Interpret the Results

The output $f^*\left(p\right)$ represents the value of the Legendre transform at slope $p$. This value can be seen as the support function of the convex function $f$ and provides a dual perspective by switching from the variable $x$ to the slope $p$.

In physics, this corresponds to changing from a coordinate-based description to a momentum-based one. In optimization, it helps analyze dual problems and understand sensitivity to changes in parameters.

Worked Example

Consider the quadratic function:

Its derivative is:

Setting $\mathrm{f\text{'}}(x)\; =\; p$ gives $\mathrm{x\; =\; p}$. Substituting back, the Legendre transform is:

This matches the well-known result for the Legendre transform of a quadratic function. Using the calculator, you can input f(x) = x^2/2 and p = 1 to verify this output.

Comparison Table: Legendre Transform vs. Related Transforms

Limitations and Assumptions

• Convexity Required: The Legendre transform is defined for convex functions. Non-convex functions may yield multiple or no solutions.
• Differentiability: The calculator assumes the function is differentiable to apply Newton's method for solving $f\text{'}(x)\; =\; p$. Non-differentiable points may cause convergence issues.
• Initial Guess Sensitivity: Newton's method requires a reasonable initial guess for $x$ to converge. Poor guesses may lead to failure or incorrect results.
• Single Variable Only: This calculator handles functions of one variable. Multivariate Legendre transforms involve gradients and Hessians, which are not supported here.
• Numerical Approximation: The method uses iterative numerical techniques and symbolic differentiation via math.js, which may introduce rounding errors or fail for complex functions.

Frequently Asked Questions

What is the purpose of the initial x guess?

The initial guess helps Newton's method start close to the solution of $f\text{'}(x)\; =\; p$. A good guess improves convergence speed and accuracy.

Can I use this calculator for non-convex functions?

The Legendre transform is not generally defined for non-convex functions. Using such functions may produce incorrect or undefined results.

Why does the calculator use Newton's method?

Newton's method efficiently solves the equation $f\text{'}(x)\; =\; p$ by iteratively improving the estimate for $x$. It requires the first and second derivatives of $f$.

Is the Legendre transform reversible?

Yes, for strictly convex and lower semicontinuous functions, applying the Legendre transform twice returns the original function.

Can I save or export the results?

This calculator provides a "Copy Result" button to copy the computed $x$ and $f^*(p)$ values for your records. CSV export is not currently supported.

How does this relate to physics?

In classical mechanics, the Legendre transform converts the Lagrangian (function of position and velocity) into the Hamiltonian (function of position and momentum), enabling analysis of conservation laws and system dynamics.