# Mapertius's Principle and 2nd Newton's law

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• MP97
MP97
Hi, I have been working on deriving the second Newton's law from the Mapertius principle applied to a perfectly elastic collision from a free fall acceleration problem. These are my calculations, but I keep getting a coefficient of 2 in my final answer for some reason. Could someone explain to me what I am doing wrong?

I appreciate any help you can provide!

## What is Mapertius's Principle?

Mapertius's Principle, also known as the Principle of Least Action, states that the path taken by a physical system between two states is the one for which the action is minimized. Action is defined as the integral of the Lagrangian (difference between kinetic and potential energy) over time. This principle is fundamental in classical mechanics and has applications in quantum mechanics and general relativity.

## How does Mapertius's Principle relate to Newton's 2nd Law?

While Newton's 2nd Law describes the relationship between force, mass, and acceleration (F = ma) in a local and instantaneous manner, Mapertius's Principle provides a global and variational perspective. Both approaches are equivalent in classical mechanics; Newton's 2nd Law can be derived from the Principle of Least Action by applying the Euler-Lagrange equations to the action integral.

## What is Newton's 2nd Law of Motion?

Newton's 2nd Law of Motion states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. Mathematically, it is expressed as F = ma, where F is the net force, m is the mass, and a is the acceleration. This law explains how the velocity of an object changes when it is subjected to an external force.

## Can Mapertius's Principle be applied to systems with non-conservative forces?

Mapertius's Principle is most straightforwardly applied to systems with conservative forces, where the potential energy can be clearly defined. For non-conservative forces, such as friction, the principle can still be applied but requires modifications. These modifications often involve incorporating additional terms to account for the energy dissipation or using generalized coordinates and constraints.

## What are some practical applications of Mapertius's Principle and Newton's 2nd Law?

Both Mapertius's Principle and Newton's 2nd Law have widespread applications in engineering, physics, and other sciences. Newton's 2nd Law is fundamental in designing mechanical systems, understanding celestial mechanics, and analyzing motion in various contexts. Mapertius's Principle is crucial in fields like optics (Fermat's Principle), quantum mechanics (path integrals), and general relativity (geodesics in spacetime). Together, they provide comprehensive tools for modeling and understanding dynamic systems.

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