The **key difference** between Lagrangian and Hamiltonian mechanics is **that Lagrangian mechanics describe the difference between kinetic and potential energies, whereas Hamiltonian mechanics describe the sum of kinetic and potential energies.**

Lagrangian mechanics and Hamiltonian mechanics are important concepts in physical chemistry that come under classical mechanics. The Lagrangian mechanics was developed by the Italian mathematician Joseph-Louis Lagrange in 1788, while the Hamiltonian mechanics was developed by William Rowan Hamilton in 1833.

### CONTENTS

1. Overview and Key Difference

2. What is Lagrangian Mechanics

3. What is Hamiltonian Mechanics

4. Side by Side Comparison – Lagrangian vs Hamiltonian Mechanics in Tabular Form

5. Summary

## What is Lagrangian Mechanics?

Lagrangian mechanics can be defined as a reformulation of classical mechanics that was introduced by the Italian mathematician named Joseph-Louis Lagrange in 1788. In this chemical concept, the trajectory of a physical system containing particles is derived through solving the Lagrange equations in one of two forms: the Lagrange equations of the first kind and the Lagrange equations of the second kind.

The first kind of Lagrange equations treats constraints explicitly as extra equations using Lagrange multipliers, while the second kind of Lagrange equations incorporates the constraints directly through judicious choice of generalized coordinates. However, in either of these two types, a mathematical function named the Lagrangian is referred to as a function of the generalized coordinates, their time derivatives, and time. Moreover, this concept contains information about the dynamics of the system.

Lagrangian mechanics is a mathematically more sophisticated and systematic chemical concept. No new physics concepts have been necessarily introduced for the application of Lagrangian mechanics when compared to Newtonian mechanics. However, Lagrangian mechanics is very useful in solving mechanical problems in physics when Newton’s formulations of classical mechanics are not convenient.

## What is Hamiltonian Mechanics?

Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics. This chemical concept contributes to the formulation of statistical mechanics and quantum mechanics. This concept was developed by William Rowan Hamilton in 1833. He developed it by starting from Lagrangian mechanics. Furthermore, Hamiltonian mechanics is equivalent to Newton’s laws of motion in the limitations of classical mechanics.

In Hamiltonian mechanics, we can use a set of canonical coordinates in describing the classical physical systems: *r* = (*q*, *p*). each of the coordinates of these components q_{i}, p_{i} is indexed to the frame of reference of that physical system. The q_{i} coordinate components are named as generalized coordinates, while p_{i} is named as their conjugate momenta.

## What is the Difference Between Lagrangian and Hamiltonian Mechanics?

Lagrangian mechanics and Hamiltonian mechanics are important concepts in physical chemistry that come under classical mechanics. Lagrangian mechanics was developed by the Italian mathematician named Joseph-Louis Lagrange in 1788, while Hamiltonian mechanics was developed by William Rowan Hamilton in 1833. The key difference between Lagrangian and Hamiltonian mechanics is that Lagrangian mechanics describe the difference between kinetic and potential energies, whereas Hamiltonian mechanics describe the sum of kinetic and potential energies. Furthermore, Lagrangian mechanics use Cartesian coordinates in calculations, whereas Hamiltonian mechanics use canonical coordinates.

Below is a summary of the difference between Lagrangian and Hamiltonian mechanics in tabular form.

## Summary – Lagrangian vs Hamiltonian Mechanics

Lagrangian mechanics can be defined as a reformulation of classical mechanics. Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics. The key difference between Lagrangian and Hamiltonian mechanics is that Lagrangian mechanics describe the difference between kinetic and potential energies, whereas Hamiltonian mechanics describe the sum of kinetic and potential energies.

##### Reference:

1. “14: Hamiltonian Mechanics.” *Physics LibreTexts*, Libretexts, 30 Dec. 2020, Available here.

##### Image Courtesy:

1. “Lagrange portrait” (Public Domain) via Commons Wikimedia

2. “WilliamRowanHamilton” By Unknown author – (Public Domain) via Commons Wikimedia

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