# Tag Info

Newtonian mechanics covers the discussion of the movement of classical bodies under the influence of forces by making use of Newton’s three laws. For more general discussion of energy, momentum conservation etc., use classical-mechanics, for Newton’s description of gravity, use newtonian-gravity.

## When to Use this Tag

Use for the discussion of classical dynamics using Newton’s three laws (cf. below). The more general topic is for the discussion of energy, (angular/linear) momentum and the study of more advanced topics.

## Introduction

Newtonian mechanics is based on three laws:

• The velocity of an object only changes if a force affects this object.
• The acceleration of an object is parallel and proportional to the net force acting on the object.
• A body exerting a force $\vec F$ on a second body experiences a force $-\vec F$ from the second body onto itself.

These laws can be expressed in two equations, with $m$ being the inertial mass of a body, $\vec F$ the force acting on the body, $\vec a$ the second time derivative of the position of said body and $\vec F_{i,j}$ the force exerted by body $i$ onto body $j$:

$$\vec F = m \vec a \quad ; \quad \vec F_{1,2} = - \vec F_{2,1}$$

Especially the first equation is often known as an equation of motion. Integrating it twice will give the trajectory of the body under the influence of the force $\vec F$.

## Alternative Formulations

Especially for complex problems with many different bodies and constraints on the motion of these bodies (such as a pendulum always being at a fixed distance from a given point), it is often difficult to find the exact force $\vec F$ acting on a particular body. Lagrangian mechanics is able to more efficiently address these problems by deriving the equations of motion from the variational principle. Hamiltonian mechanics extends this concept by applying Legendre transformations to reach a coordinate system where the energy of a body is zero; hence making it easy to solve the equations of motion.