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Newtonian Mechanics

Newtonian Mechanics is one of the most elemental frameworks for understanding the interplay between motion and forces via ordinary differential equations. In fact, modeling motion via ordinary differential equations directly leads to Newton’s three laws of motion.
Being able to wield the tools and techniques that underlie Newtonian Mechanics is crucial for aerospace engineers as it allows us to analyze and predict the behavior of aircraft and spacecraft under various conditions. The design, stability, and control of aerospace vehicles, whether in atmospheric flight or in space, are fundamentally governed by the same laws of motion that Newton proposed in the 17th century.

Heads up!

Newtonian mechanics does have its limitations. In scenarios involving extremely high speeds (close to the speed of light) or when dealing with very small scales where quantum effects become relevant, other approaches need to be considered.

Newton’s Laws of Motion

Newton’s First Law (Inertia):
This law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
Newton’s Second Law ():
The second law states that an external force is necessary to change the momentum () of an object.

This law defines the relationship between force, mass, and acceleration.

Newton’s Third Law (Action–Reaction):
This law states that for every action, there is an equal and opposite reaction. This principle is directly related to propulsion systems, such as jet engines and rocket thrusters, which rely on expelling mass in one direction to produce thrust in the opposite direction.

Heads up!

is and approximation that holds iff !

Equations of motion

The aim of Newtonian Mechanics is it to generate so-called “equations of motion”, i.e. sets of differential equations that, when solved describe the trajectories of objects of interest. Depending on the problem those equations can take the form of coupled systems of ordinary differential equations. In the simplest cases where we are interested in the dynamics of one point-like particle equations of motion may be derived via

Newtonian Equations of Motion

Here, is the momentum of the object under investigation, and are external forces acting on the same object.

Example: 3D Harmonic Oscillator

Assuming , and , where is the mass of the oscillator, is a constant and the displacement with respect to the equilibrium position, the equations of motion can be derived via

which results in

The above equation corresponds to three sets of second order ordinary differential equations, that even permit an analytic solution of the form , with being constants of integration and .

Integrals of Motion in Newtonian Dynamics

The concept of integrals of motion refers to quantities that remain constant as a system evolves over time. Emmy Noether (1882–1935) pioneered the idea that conserved quantities such as energy or angular momentum are a consequence of symmetries in the system.

The term “ten integrals of motion” typically refers to the ten independent conserved quantities that arise in the most general mechanical system with a fixed number of particles in three-dimensional space. These integrals fall into three main categories:

Linear Motion (Translational Symmetries): Six integrals describe the conservation of linear and angular momentum for the entire system.
Angular Motion (Rotational Symmetries): Three integrals describe the orientation or rotational motion of the system, in particular the length and orientation of the system’s angular momentum.
Mechanical Energy: The conservation of mechanical energy is linked to the time-reversibility of a system. Friction is a counter-example, because the arrow of time determines whether mechanical energy is converted into heat (internal energy) or vice versa.

Each integral of motion corresponds to a physical law or symmetry in the system. These conserved quantities allow engineers and physicists to simplify the analysis of complex dynamical systems, such as the flight trajectory of an aircraft or the orbit of a satellite. Understanding these integrals is particularly important in systems with minimal external forces, where such quantities remain conserved over time.

Linear Momentum (3 Integrals)

Linear momentum is conserved in a system where there are no external forces acting. For a particle or system of particles, the three components of linear momentum in the -, -, and -directions are:

These correspond to the translational symmetry of space, meaning the laws of physics do not change if the system is displaced in any direction.

Angular Momentum (3 Integrals)

Angular momentum is conserved in a system with no external torques. The three components of angular momentum about the -, -, and -axes are:

These correspond to the rotational symmetry of space, implying that the physics remains the same if the system is rotated.

Energy (1 Integral)

The total mechanical energy (kinetic plus potential) of a system is conserved in the absence of non-conservative forces (like friction or air resistance). The conservation of mechanical energy can be expressed as:

where is the kinetic energy and is the potential energy. Energy conservation corresponds to the time-invariance of the system—if the system does not explicitly depend on time, energy is conserved.

Center of Mass Motion (3 Integrals)

The motion of the center of mass of a system of particles follows the same laws of motion as a single particle. The position of the center of mass in the -, -, and -directions can be described by three integrals of motion. For a system of particles with masses and positions , the center of mass is given by:

where is the total mass of the system.