Newton's Second Law

Prerequisites: Calculus.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• Definitions

Intuitive

Newton's second law tells us that how an object $\color{magenta}{\text{gets faster}}$ depends on its $\color{olive}{\text{mass}}$ and the $\color{blue}{\text{total force }}$ acting on it. The total force acting on it depends on the $\color{red}{\text{location of the object.}}$

Formulated a bit differently, it tells us that the $\color{magenta}{\text{acceleration}}$ of an object is given by the ratio of the $\color{blue}{\text{force }}$ acting on it and its $\color{olive}{\text{mass}}$.

The acceleration is the rate of change of the velocity. The velocity is the rate of change of the location.

However, take note that it is not sufficient to describe a physical system. Additionally, to describe a system we need to know what forces act on the object and what equations describe them. Famous examples of such force laws are

• Newton's law of gravity
• Lorentz' force law
• Coulomb's force law

So, for example, when we want to describe the movement of a planet around the sun we need to think about what forces act on the planet. For this system gravity is the most important force since both objects - the sun and the planet - are very heavy. Therefore, to calculate the movement of the planet, all we have to do is use Newton's law of gravity to calculate the force acting on it. Then, when we have calculated the force we can use Newton's second law to calculate the acceleration of the object. Then, given some starting point and starting velocity of the planet we can calculate where the planet will be at every point in time in the future.

Concrete

Boundary or Initial Conditions

Since Newton's second law contains the second derivative of the location: $\vec a = \frac{\partial^2 r}{\partial t^2}$, we need two boundary conditions to solve it. Whenever the independent variable is time, boundary conditions are usually called initial conditions. (In technical terms, we say that Newton's law is a second order differential equation.) For example, we can use the location of the object at the starting time and the velocity at the starting time as boundary conditions. Alternatively, we could use, for example, the location of the object at two different points in time as boundary conditions.

Example: The movement of an asteroid towards the sun

As a first step, we use Newton's law of gravity to calculate the force acting on the asteroid

$$F = G \frac{m_s m_a}{r^2},$$

where $G$ is the gravitational constant, $m_s$ the mass of the sun, $m_a$ the mass of the asteroid and $r$ the distance between them. The force points towards the sun, so along the radial direction it enters with a minus sign. We assume that the asteroid is much lighter than the sun and therefore can neglect the effect the asteroid has on the sun. In other words, we assume the sun remains stationary, although of course an equal gravitational force also acts on the sun.

We can then put this equation into Newton's second law

$$F = m_a a \quad \rightarrow \quad -G \frac{m_s m_a}{r^2} = m_a a ,$$

Next, we recall that the acceleration $a$ is just the rate of change of the velocity and subsequently that the velocity is just the rate of change of the location:

$$a = \dot v = \ddot r .$$

Putting this into our equation yields a differential equation for the location $r$ that we need to solve

$$-G \frac{m_s m_a}{r^2} = m_a \ddot r .$$

The equation is not easy to solve, and a nice discussion can be found here.

Take note that before we can fully solve the equation, we need to specify the boundary conditions.

• No-Nonsense Classical Mechanics by Jakob Schwichtenberg, chapter "Newtonian Mechanics"

Abstract

For $N$ point particles, the configuration space is $Q = \mathbb R^{3N}$ and Newton's second law is the system of second-order ordinary differential equations

$$m_k\,\ddot{\vec r}_k = \vec F_k(\vec r_1,\dots,\vec r_N,\dot{\vec r}_1,\dots,\dot{\vec r}_N,t), \qquad k = 1,\dots,N .$$

Writing $v = \dot q$, this becomes a first-order system on the tangent bundle $TQ$ (the space of positions and velocities): the law defines a vector field $X$ on $TQ$ of a special kind, a second-order vector field, whose integral curves are the lifts $t\mapsto(q(t),\dot q(t))$ of curves in $Q$. If $F$ is locally Lipschitz, the Picard-Lindelöf theorem gives a unique local solution for each initial position and velocity, which is the mathematical content of determinism in Newtonian mechanics. Without the Lipschitz condition uniqueness can fail, as in Norton's dome.

More generally, let $Q$ be a manifold with the Riemannian metric $g$ defined by the kinetic energy, $T = \frac12 g(\dot q,\dot q)$. For holonomic constraints this is the induced metric weighted by the masses. Newton's law then takes the coordinate-free form

$$\nabla_{\dot q}\,\dot q = F^\sharp ,$$

where $\nabla$ is the Levi-Civita connection of $g$ and $F^\sharp$ is the force one-form (the generalized force) turned into a vector with $g$. Free motion is geodesic motion. For a conservative force, $F = -dV$, the equation is the Euler-Lagrange equation of $L = T - V$ (see the Lagrangian formalism), and the flow preserves the energy $T + V$.

The law holds in inertial frames. Galilean spacetime is an affine space with an absolute time function and a Euclidean structure on the hypersurfaces of constant time. The Galilei group, generated by rotations, boosts $\vec r\mapsto\vec r + \vec v t$, translations in space and time, maps inertial frames to inertial frames. Boosts and translations leave $\ddot{\vec r}$ unchanged, and rotations rotate it. The law is consistent with Galilean relativity when the force law is Galilei covariant, for example when the forces depend only on relative positions and relative velocities of the particles and transform as vectors under rotations. In Newton-Cartan theory, gravity is absorbed into a connection on Galilean spacetime, in the same way that free fall becomes geodesic motion in general relativity.

Why is it interesting?

Newton's second law is the most fundamental equation of classical mechanics. It is still used nowadays, for example, by engineers.

I can't think of another equation that has had more dramatic impact on our history and our world. So much of what we have built around us started with knowing $F=ma$.Rob Moore

Definitions

• $\vec F$ is the force that acts on the object in question.
• $\vec a$ is the acceleration of the object. The acceleration is the rate of change of the velocity $\vec v$ of the object, which in turn is the rate of change of the location $\vec r$ of the object: $a = \ddot{r} = \dot{v}$.

Leads to: Harmonic Oscillator, Lorentz Force Law, Newton's law of Gravity, Newtonian Mechanics, The Navier-Stokes Equations, Work-Energy theorem.