Tag Info

Newtonian mechanics discusses the movement of classical bodies under the influence of forces by applying Newton’s three laws. For more general concepts, use [classical-mechanics]. For Newton’s description of gravity, use [newtonian-gravity].

When to Use This Tag

Use for discussing classical dynamics using Newton's three laws (cf. below). The more general topic is for discussions on more advanced topics.

## Introduction

The two main ingredients of Newtonian mechanics are the concepts of trajectories and forces. The former is described by specifying the path of the body as a function of time, and from which one can define its and by taking derivatives. The latter affects and determines the former using the following three laws:

• The velocity of an object only changes if and only 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 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 $$j$$:

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

The first equation is often known as an equation of motion and is a second-order ordinary differential equation. Under some general assumption on the force $$\vec F$$, and after specifying some initial conditions, this equation has a unique solution. This solution is 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 can more efficiently address these problems by deriving the equations of motion from a variational principle. Hamiltonian mechanics extends this concept by applying Legendre transformations which makes some features of the formalism more transparent, such as symmetries and canonical transformations. The Hamiltonian formulation is also convenient when one is interested in numerically integrating the equations of motion since they become a system of first-order equations instead of second-order.