Newtonian Motion Models

Newton’s Laws of Motion

Newton’s Laws of Motion are the foundation of classical (Newtonian) mechanics. They describe the relationship between the forces acting on a body and the motion of that body.

First Law - Inertia

A body at rest will remain at rest, and a body in uniform motion will continue in uniform motion unless acted upon by an external force.

This law defines the concept of inertia: objects resist changes to their velocity unless a net external force is applied.

Second Law - Force and Acceleration

At any moment in time, the total (net) force acting on a body is equal to the product of its mass and its acceleration:

$$ \vec{F_{\text{net}}} = m \vec{a} $$

Where:

• $\vec{F_{\text{net}}}$ is the total force vector acting on a body

This is the central equation governing motion in Newtonian mechanics and serves as the basis for numerical integration over discrete time steps.

Alternatively, the second law can be described as the rate of change of the body's momentum over time. Expressed as:

$$ \vec{F_{\text{net}}} = \frac{\Delta p}{\Delta t} $$

$$ p = m v $$

This form is especially useful when dealing with systems where mass is not constant or when expressing impulse-based interactions.

Third Law — Action and Reaction

For every action, there is an equal and opposite reaction. If body 1 exerts a force on body 2, then body 2 simultaneously exerts a force of equal magnitude and opposite direction on body 1:

$$ \vec{F_{12}} = -\vec{F_{21}} $$

Where:

• $\vec{F_{ij}}$ is the force exerted on body $i$ by body $j$.

This principle ensures that interactions between bodies conserve momentum and supports the symmetric treatment of forces in both analytical derivations and numerical computation.

Force Accumulation and Superposition

In Newtonian mechanics, the net force acting on a body is the vector sum of all individual forces applied to it. This principle is known as the superposition of forces.

If multiple forces $( \vec{F}_1, \vec{F}_2, \dots, \vec{F}_n )$ act on a single body, the total (net) force is given by:

$$ \vec{F_{\text{total}}} = \sum_{i=1}^{N}\vec{F_i} $$

This superposition principle applies to all force types, including but not limited to:

• Gravitational forces
• Contact forces (normal, frictional)
• Applied or external forces
• Electromagnetic forces (if present)

Each force contributes independently to the net force at each time-step, and their vector sum determines the resulting acceleration of the body through Newton’s Second Law:

$$ \vec{F}_{\text{net}} = m \vec{a} $$

This additive behavior is fundamental to all multi-force systems and provides the basis for both analytical formulations and computational models in classical mechanics.

Kinematics: Position, Velocity, Acceleration

Kinematics describes the motion of bodies without reference to the forces that cause it. In Newtonian mechanics, the motion of a body is characterized by three time-dependent vector quantities:

• Position $\vec{x}_{t}$
• Velocity $\vec{v}_{t}$
• Acceleration $\vec{a}_{t}$

Velocity

Velocity is the time derivative of position:

$$ \vec{v}_{t} = \frac{d\vec{x}_{t}}{dt} $$

Acceleration

Acceleration is the time derivative of velocity and, thus, the second time derivative of the position:

$$ \vec{a}_{t} = \frac{d\vec{v}_{t}}{dt} = \frac{d^2\vec{x}_{t}}{dt^2} $$

Numerical Integration of Motion

In physics simulations, motion is computed step-by-step using discrete time intervals (timesteps). Because acceleration, velocity, and position are continuous quantities defined by differential equations, we use numerical integration methods to approximate their values over time.

This section introduces common numerical integration methods used to simulate motion in Newtonian mechanics.

Euler Integration (Explicit)

Euler integration is the simplest numerical method. It updates velocity and position using the current acceleration and velocity values:

$$ \vec{v}_{t + \Delta t} = \vec{v}_t + \vec{a}_t \Delta t $$

$$ \vec{x}_{t + \Delta t} = \vec{x}_t + \vec{v}_t \Delta t $$

Advantages:

• Easy to implement
• Computationally inexpensive

Disadvantages:

• Can be unstable if the timestep is too large
• Errors accumulate quickly due to linear approximation

Euler integration is suitable for simple or preliminary simulations but is generally less accurate for complex or long-running simulations.

Verlet Integration

Verlet integration emphasizes updating position directly and implicitly incorporates velocity by referencing previous positions and acceleration:

$$ \vec{x}_{t + \Delta t} = 2\vec{x}_{t} - \vec{x}_{t} + \vec{a}_t \Delta t^2 + \mathcal{O}(\Delta t^4) $$

Advantages:

• Time-reversible and symplectic, which helps conserve energy
• Very stable for systems with conservative forces (e.g., orbital simulations, particle systems)

Disadvantages:

• Does not explicitly track velocity (though it can be approximated)
• Requires storing previous positions, which complicates initialization
• Assumes a constant time-step

Verlet integration is widely used in molecular dynamics and particle systems for its stability and energy conservation.

Velocity Verlet Integration

Velocity Verlet combines the benefits of Verlet’s stability with explicit velocity tracking:

$$ \vec{x}_{t + \Delta t} = \vec{x}_{t} + \vec{v}_{t} \Delta t + \frac{1}{2} \vec{a}_{t} \Delta t^2 $$

$$ \vec{v}_{t + \Delta t} = \vec{v}_{t} + \frac{\vec{a}_{t} + \vec{a}_{t + \Delta t}}{2} \Delta t $$

Advantages:

• Combines accuracy with stability
• Explicit velocity simplifies force and momentum calculations
• Commonly used in physics engines and molecular simulations

Disadvantages:

• Slightly more complex than basic Verlet or Euler methods
• Assumes a constant time-step

Velocity Verlet is a solid, practical choice for most physics simulations requiring both stable and accurate integration.

Future Work: Runge-Kutta Methods

Runge-Kutta methods, especially the classical 4th order Runge-Kutta (RK4), provide even higher accuracy by sampling multiple intermediate slopes within each timestep.

• More computationally intensive but significantly more precise
• Useful for simulations with rapidly changing or stiff forces
• A candidate for future improvements to increase accuracy in your simulations

While not covered here, Runge-Kutta methods offer a tradeoff of greater accuracy for higher computational cost, and are useful for complex or stiff systems.

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