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1.1 Newtonian Mechanics

We begin our study of theoretical physics with a review of Newtonian mechanics. Although in modern physics we consider Newtonian mechanics to be superseded by more accurate theories such as special relativity and quantum mechanics, it remains an essential foundation for understanding the physical world.

Table of Contents​

Single-Particle Mechanics​

The position of a particle in three-dimensional space can be described by a vector function of time, where the vector lives within . The velocity is the time derivative of the position vector, and the linear momentum is defined as the product of mass and velocity. As we have seen these concepts before, we will now formalize them with precise definitions.

Definition 1.1.1 Velocity

The velocity of a particle is defined as the time derivative of its position vector ;

Definition 1.1.2 Linear Momentum

The linear momentum of a particle with mass and velocity is defined as

Definition 1.1.3 Force

The force acting on a particle is defined as the time derivative of its linear momentum ;

Newton's Second Law of Motion states that the force acting on a particle is equal to the time derivative of its linear momentum. For a particle with constant mass, this can be simplified to

where is the acceleration of the particle.

Definition 1.1.4 Acceleration

The acceleration of a particle is defined as the time derivative of its velocity ;

Definition 1.1.5 Reference Frame

A reference frame is a coordinate system used to describe the position and motion of objects. It consists of an origin and a set of axes that define the spatial dimensions. A reference frame can be inertial or non-inertial.

In Newtonian mechanics, an inertial reference frame is one in which Newton's laws of motion hold true. This means that a particle not subjected to any net external force will either remain at rest or continue to move at a constant velocity in a straight line.

Theorem 1.1.6 Conservation of Linear Momentum for a Particle

In an inertial reference frame, if no external force acts on a particle, its linear momentum remains constant over time, so .

Definition 1.1.7 Angular Momentum

The angular momentum of a particle with position vector and linear momentum is defined as

Definition 1.1.8 Torque

The torque is the cross product of the position vector and the force acting on a particle;

Theorem 1.1.9 Torque is the Time Derivative of Angular Momentum

In an inertial reference frame, the torque acting on a particle is equal to the time derivative of its angular momentum;

Proof.

Using the definitions of angular momentum and torque, we have

Thus, we have shown that .

Theorem 1.1.10 Conservation of Angular Momentum for a Particle

In an inertial reference frame, if no external torque acts on a particle, its angular momentum remains constant over time, so .

Energy and Work​

Definition 1.1.11 Work

Suppose a force acts on a particle that moves from position to position along a path . The work done by the force on the particle is defined as the line integral

Definition 1.1.12 Kinetic Energy

The kinetic energy of a particle with mass and velocity is defined as

where is the magnitude of the velocity.

Theorem 1.1.13 Work-Energy Theorem

The work done by the net force acting on a particle with constant mass as it moves from position to position is equal to the change in its kinetic energy;

where and are the kinetic energies at positions and , respectively.

Proof.

From the definition of work, we have

Using Newton's Second Law, we can substitute into the integral, yielding

Finally, we recognize that , allowing us to rewrite the integral as

Thus, we have shown that .

Definition 1.1.14 Force Field

A force field is a vector field that assigns a force vector to every point in space.

Definition 1.1.15 Conservative Force Field

A force field is said to be conservative if the work done by the force on a particle moving between any two points is independent of the path taken.

Theorem 1.1.16 Equivalent Conditions for Conservative Force Fields

For a force field defined on a simply connected domain in , the following conditions are equivalent:

  1. The work done by the force on a particle moving between any two points is independent of the path taken.
  2. The line integral of the force around any closed loop is zero; that is, for any closed curve .
  3. There exists a scalar potential function such that . This function is called the potential energy associated with the force field.

Proof.

(1) (2): If the work done is independent of the path, then for any closed loop , the work done must be zero, since the starting and ending points are the same.

(2) (3): If the line integral around any closed loop is zero, we can define a scalar potential function by choosing a reference point and defining

As the line integral is path-independent, is well-defined. Taking the gradient of gives .

(3) (1): If there exists a scalar potential function such that , then the work done by the force when moving from point to point is

which depends only on the endpoints and , and not on the path taken between them. Thus, the work done is independent of the path taken.

Thus, we have shown the equivalence of the three conditions.

As the potential energy is the negative of the work done by the force, the force is in turn the negative gradient of the potential energy:

Theorem 1.1.17 Conservation of Mechanical Energy

In an inertial reference frame, for a particle moving under the influence of a conservative force field, the total mechanical energy , defined as the sum of kinetic energy and potential energy , remains constant over time.

Proof. Let the particle move from position to position under the influence of a conservative force field. From the Work-Energy Theorem, we have

where and are the kinetic energies at positions and , respectively.

Since the force is conservative, the work done by the force is equal to the negative change in potential energy:

where and are the potential energies at positions and , respectively.

Equating the two expressions for work, we have

Thus we have shown that the total mechanical energy remains constant over time.

Systems of Particles​

In the case of a system of multiple particles, we can extend the definitions and theorems from single-particle mechanics to account for interactions between particles.

Newton's second law for a system of particles can be split into external and internal forces. The external force acts on particle from outside the system, while the internal force is the force exerted on particle by particle within the system. Thus, the total force acting on particle is given by

Newton's second law can thus be stated as

Definition 1.1.18 Law of Action and Reaction

The weak law of action and reaction, or Newton's Third Law, states that for every action, there is an equal and opposite reaction. In the context of a system of particles, this means that the internal forces between any two particles are equal in magnitude and opposite in direction;

The full action-reaction law also requires that the forces act along the line connecting the two particles. Note that the weak law is not always satisfied in certain physical situations, such as in electromagnetic interactions where forces can be non-central.

Summing equation over all particles in the system, we obtain

Using the weak law of action and reaction, the double sum on the right-hand side vanishes, leaving us with

If we define a center of mass for the system of particles as following:

Definition 1.1.19 Center of Mass

The center of mass of a system of particles with masses and position vectors is defined as a weighted average of their positions;

where is the total mass of the system.

Then we can rewrite the equation as

Definition 1.1.20 Total Linear Momentum

The total linear momentum of a system of particles is defined as the vector sum of the linear momenta of all the particles;

Theorem 1.1.21 Conservation of Total Linear Momentum for a System of Particles

In an inertial reference frame, if no external force acts on a system of particles, the total linear momentum of the system remains constant over time, so .

Definition 1.1.22 Total Angular Momentum

The total angular momentum of a system of particles with respect to a chosen origin is defined as the vector sum of the angular momenta of all the particles;

Definition 1.1.23 Total Torque

The total torque acting on a system of particles with respect to a chosen origin is defined as the vector sum of the torques acting on all the particles;

We can expand the total torque as follows:

If the internal forces are central forces, meaning they act along the line connecting particles and (thus obeying the full action-reaction law), then the second term vanishes, leaving us with

In other words, internal forces do not contribute to the total torque when the full action-reaction law is satisfied.

Theorem 1.1.24 Conservation of Total Angular Momentum for a System of Particles

In an inertial reference frame, if no external torque acts on a system of particles, the total angular momentum of the system remains constant over time, so .

To further analyze the motion of a system of particles, note that the total angular momentum is a combination of the angular momentum due to the motion of the center of mass and the angular momentum due to the motion of the particles relative to the center of mass. To see this, we define as the position vector of particle relative to the center of mass . Then we have the following theorem:

Theorem 1.1.25 Decomposition of Total Angular Momentum

The total angular momentum of a system of particles can be decomposed into the angular momentum due to the motion of the center of mass and the angular momentum due to the motion of the particles relative to the center of mass;

Proof. We obviously have

where is the velocity of the center of mass, and is the velocity of particle relative to the center of mass.

The total angular momentum can thus be decomposed as follows:

Noting that by the definition of the center of mass, we have

The first term represents the angular momentum due to the motion of the center of mass, while the second term represents the angular momentum due to the motion of the particles relative to the center of mass. This means that depends on the choice of origin for .

Next, we can analyze the work and energy for a system of particles. Suppose the system of particles moves from configuration to configuration under the influence of external forces and internal forces . The total work done on the system is given by

Using Newton's second law for each particle and summing over all particles, we find that the total work done is equal to the change in the total kinetic energy of the system:

where and are the total kinetic energies of the system at configurations and , respectively, defined as follows:

Definition 1.1.26 Total Kinetic Energy

The total kinetic energy of a system of particles is defined as the sum of the kinetic energies of all the particles;

Theorem 1.1.27 Work-Energy Theorem for a System of Particles

The total work done by the external and internal forces on a system of particles as it moves from configuration to configuration is equal to the change in the total kinetic energy of the system;

where and are the total kinetic energies at configurations and , respectively.

We can also use the center-of-mass coordinates and to decompose the total kinetic energy into the kinetic energy due to the motion of the center of mass and the kinetic energy due to the motion of the particles relative to the center of mass:

Theorem 1.1.28 Decomposition of Total Kinetic Energy

The total kinetic energy of a system of particles can be decomposed into the kinetic energy due to the motion of the center of mass and the kinetic energy due to the motion of the particles relative to the center of mass;

Proof. Expanding the total kinetic energy, we have

Noting that by the definition of the center of mass, we have

The first term represents the kinetic energy due to the motion of the center of mass, while the second term represents the kinetic energy due to the motion of the particles relative to the center of mass.

If the external forces are conservative, we can define a potential energy function for the internal forces, such that the first term in can be rewritten as

where indicates a gradient with respect to the position vector of particle , and is the potential energy associated with the external force acting on particle . If the internal forces are also conservative, we can define a potential energy function for the internal forces, denoted . Note that the strong law of action and reaction means that must only depend on the distance between particles and :

Theorem 1.1.29 Strong Law of Action and Reaction and Internal Forces

If the strong law of action and reaction holds for a conservative internal force between particles and , then the internal force automatically obeys Newton's Third Law, and the force pairs lie along the line connecting the two particles.

Proof. By definition, the internal force is related to the potential energy function by

Thus, we have shown that . Furthermore, since depends only on the distance between particles and , the gradient points along the line connecting the two particles, ensuring that the forces are central.

With this in mind, the second term of can be rewritten as

If we define , and as the gradient with respect to (i.e., ), then we have

The half factor is included to avoid double counting the potential energy between each pair of particles. Therefore, combining both terms, we can define a total potential energy for the system as follows:

Definition 1.1.30 Total Potential Energy

The total potential energy of a system of particles is defined as the sum of the potential energies associated with the external forces acting on each particle and the potential energies associated with the internal forces between each pair of particles, provided that the internal forces are conservative and obey the strong law of action and reaction;

The first term represents the total potential energy due to external forces, while the second term represents the total potential energy due to internal forces.

Definition 1.1.31 Rigid Body

A rigid body is a system of particles in which the distances between all pairs of particles remain constant over time, regardless of the external forces acting on the system. This means that the shape and size of the rigid body do not change during its motion.

Rigid bodies can undergo translational and rotational motion, but the relative positions of the particles within the body remain fixed. Thus, their internal potential energy remains constant, and any changes in the total energy of the system are due to changes in kinetic energy or external potential energy.

Theorem 1.1.32 Conservation of Mechanical Energy for a System of Particles

In an inertial reference frame, for a system of particles moving under the influence of conservative external and internal forces, the total mechanical energy , defined as the sum of total kinetic energy and total potential energy , remains constant over time;

Summary and Next Steps​

In this section, we have established the fundamental principles of Newtonian mechanics for both single particles and systems of particles. We defined key concepts such as force, momentum, energy, and reference frames, and we derived important theorems including Newton's laws of motion, conservation of momentum, and conservation of energy.

Here are the key takeaways:

  • Newton's Second Law relates the force acting on a particle to its rate of change of momentum.
  • Conservation laws for linear and angular momentum hold in inertial reference frames.
  • Work done by forces leads to changes in kinetic energy, as described by the Work-Energy Theorem.
  • Conservative forces allow for the definition of potential energy, leading to the conservation of mechanical energy.
  • Systems of particles can be analyzed by separating external and internal forces, with the center of mass playing a crucial role in simplifying the dynamics.

With these foundations in place, the next section will explore d'Alembert's Principle and the formulation of Lagrangian mechanics, which provides a powerful framework for analyzing the dynamics of more complex systems, including those with constraints. This will pave the way for a deeper understanding of classical mechanics, which is important for our later discussions of field theory.