---
title: "Optical Invariant (Lagrange) calculator — AperiOptic"
description: "Conserved Lagrange invariant of a system. Free online optics calculator, with the formula and code snippets."
url: "https://aperioptic.com/calculators/optical-invariant-lagrange"
---

# Optical Invariant (Lagrange) calculator

Conserved Lagrange invariant of a system.

Category: [Imaging](https://aperioptic.com/calculators)

Formula: `H = n·y·u' - n·y'·u`

## Inputs

| Input | Parameter | Unit | Default |
| --- | --- | --- | --- |
| Index n | `n` |  | 1 |
| Marg. ray y | `y` | mm | 5 |
| Chief u' | `up` | rad | 0.1 |
| Chief y' | `yp` | mm | 2 |
| Marg. u | `u` | rad | 0.05 |

## Result with the default inputs

- Invariant H: 0.4 mm·rad

Open it with your own values: put each input in the address as a parameter, for example [`/calculators/optical-invariant-lagrange?n=1`](https://aperioptic.com/calculators/optical-invariant-lagrange?n=1). A unit other than the listed one goes in `<parameter>_unit`.

[All optics calculators](https://aperioptic.com/calculators)

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The interactive page: https://aperioptic.com/calculators/optical-invariant-lagrange (AperiOptic runs in the browser; free, no account). Every page of the site as Markdown: https://aperioptic.com/llms.txt