Modern Theoretical Physics From Classical Mechanics to Quantum Field Theory
Topic

Newton's Laws of Motion

Classical Mechanics Foundations

Overview

Newton’s three laws of motion form the foundation of classical mechanics. They describe how objects move under the influence of forces and establish the framework that all later formulations — Lagrangian, Hamiltonian, and ultimately quantum mechanics — build upon.

For a reader with a mathematical background, Newton’s laws are best understood as a system of ordinary differential equations on \mathbb{R}^3, with forces as the driving terms.

The Three Laws

First Law (Inertia)

A body remains at rest or in uniform straight-line motion unless acted upon by a net external force.

Mathematically: if \mathbf{F} = 0, then

\frac{d\mathbf{v}}{dt} = 0 \quad \Longrightarrow \quad \mathbf{v}(t) = \mathbf{v}_0 = \text{const.}

This law is more subtle than it appears — it defines the existence of inertial reference frames, i.e., coordinate systems in which the law holds. The mathematical content is that such frames exist and are related by the Galilean group of transformations:

\mathbf{x}’ = R\mathbf{x} + \mathbf{v}t + \mathbf{a}, \quad t’ = t + s

where R \in O(3), \mathbf{v}, \mathbf{a} \in \mathbb{R}^3, and s \in \mathbb{R}.

Second Law (F = ma)

The rate of change of momentum equals the applied force.

\mathbf{F} = \frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt}

For constant mass, this becomes the fundamental equation of Newtonian mechanics:

\mathbf{F} = m\mathbf{a} = m\frac{d^2\mathbf{x}}{dt^2}

This is a second-order ODE for the trajectory \mathbf{x}(t) \in \mathbb{R}^3. Given a force law \mathbf{F}(\mathbf{x}, \dot{\mathbf{x}}, t) and initial conditions (\mathbf{x}_0, \mathbf{v}_0), the Picard-Lindelöf theorem guarantees a unique local solution (provided \mathbf{F} is Lipschitz continuous).

Example: Free fall near Earth’s surface

m\ddot{y} = -mg \quad \Longrightarrow \quad \ddot{y} = -g

Solution:

y(t) = y_0 + v_0 t - \frac{1}{2}g t^2

Example: One-dimensional motion in a potential

If F = -V’(x) (force derived from a potential), the equation of motion is:

m\ddot{x} = -\frac{dV}{dx}

This structure — force as the negative gradient of a potential — is the starting point for the Lagrangian formulation.

Third Law (Action and Reaction)

For every force that body A exerts on body B, body B exerts an equal and opposite force on body A.

\mathbf{F}{A \to B} = -\mathbf{F}{B \to A}

This law has an important mathematical consequence: it guarantees conservation of total momentum for a closed system. For two particles:

\frac{d}{dt}(\mathbf{p}1 + \mathbf{p}_2) = \mathbf{F}{2 \to 1} + \mathbf{F}_{1 \to 2} = 0

More generally, for N particles with pairwise forces satisfying the third law, the total momentum \mathbf{P} = \sum_{i=1}^{N} m_i \dot{\mathbf{x}}_i is conserved.

Systems of Particles

For N particles, Newton’s second law gives a system of 3N coupled second-order ODEs:

m_i \ddot{\mathbf{x}}i = \mathbf{F}_i^{\text{ext}} + \sum{j \neq i} \mathbf{F}_{ji}, \quad i = 1, \ldots, N

The configuration space is \mathbb{R}^{3N}, and the state of the system at any time is a point in the phase space \mathbb{R}^{6N} (positions and velocities).

Center of Mass

The center of mass \mathbf{X} = \frac{1}{M}\sum_i m_i \mathbf{x}_i (with M = \sum_i m_i) obeys:

M\ddot{\mathbf{X}} = \mathbf{F}^{\text{ext}}_{\text{total}}

The internal forces cancel by Newton’s third law. This reduces the problem: the center of mass moves as if it were a single particle.

Forces in Newtonian Mechanics

Gravity

Newton’s law of gravitation between two masses:

\mathbf{F} = -\frac{Gm_1 m_2}{|\mathbf{x}1 - \mathbf{x}_2|^2}\hat{\mathbf{r}}{12}

This is a conservative force derivable from the potential:

V(r) = -\frac{Gm_1 m_2}{r}

Spring Force (Hooke’s Law)

F = -kx

This leads to the harmonic oscillator — see the dedicated topic The Harmonic Oscillator for a thorough treatment.

Energy Conservation

For a particle in a potential V(\mathbf{x}), define the total energy:

E = \frac{1}{2}m|\dot{\mathbf{x}}|^2 + V(\mathbf{x})

Then:

\frac{dE}{dt} = m\dot{\mathbf{x}} \cdot \ddot{\mathbf{x}} + \nabla V \cdot \dot{\mathbf{x}} = \dot{\mathbf{x}} \cdot (m\ddot{\mathbf{x}} + \nabla V) = 0

since m\ddot{\mathbf{x}} = -\nabla V. Energy is conserved for conservative forces.

Limitations and Beyond

Newton’s laws work extraordinarily well for everyday scales, but they have limitations:

  1. Constrained systems: For particles on surfaces or curves, constraint forces are awkward to handle. The Lagrangian formulation handles constraints naturally via generalized coordinates.

  2. Non-inertial frames: Fictitious forces (Coriolis, centrifugal) must be added. The Lagrangian approach treats all frames uniformly.

  3. Many degrees of freedom: The vector equation \mathbf{F} = m\mathbf{a} becomes unwieldy. Lagrangian and Hamiltonian methods provide systematic tools.

  4. Connection to quantum mechanics: There is no obvious path from \mathbf{F} = m\mathbf{a} to quantum mechanics. The Hamiltonian formulation leads directly to canonical quantization.

These motivations lead to the Lagrangian and Hamiltonian Formalism, which reformulates classical mechanics in a way that generalizes to field theory and quantum mechanics.

References

  • I. Newton, Philosophiæ Naturalis Principia Mathematica (1687)
  • L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Pergamon, 1976), Chapter 1
  • V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, 1989), Part I
  • H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison-Wesley, 2002), Chapter 1
  • D. Tong, Lectures on Classical Dynamics — Free online notes