D1. Foundational laws of Newtonian dynamics
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Dynamics is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.
The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”
There are two types of problems in dynamics, depending on the data and the unknowns:
- Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.
- Unknown force: based on a predetermined movement (i.e., given the evolution of the system's coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.
As any scientific theory, Newton's mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton's main work (Philosophiæ Naturalis Principia Mathematica), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.
In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the Principia Mathematica (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in Rigid Body Dynamics, Batlle&Barjau, chapter 1.
D1.1 Galilean reference frames
The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:
[math]\displaystyle{ \text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces}) }[/math]
This equation shows that, since the motion of [math]\displaystyle{ \Ps }[/math] depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R.
The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of [math]\displaystyle{ \Ps }[/math] is not only related to the existence of material objects (as intuition says). A simple example makes this clear.
Let us consider a small object (a particle [math]\displaystyle{ \Ps }[/math]) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.
From an initial time instant [math]\displaystyle{ t_0 }[/math] in which the particle is at rest with respect to the surface [math]\displaystyle{ (\vel{P,$\ts_0$}{R} = \vec{0}) }[/math] and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, [math]\displaystyle{ \vel{P}{R} }[/math], Figure D1.1).
Different evolutions can be seen, depending on the movement of the surface relative to the ground:
- If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle [math]\displaystyle{ \Ps }[/math] does not move relative to the wagon, and the springs maintain their length (Figure D1.2). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, the space of that reference frame is homogeneous and isotropic, and time is uniform. Therefore, the reference frame does not play any part in the result.
- If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, [math]\displaystyle{ \Ps }[/math] initially moves in the longitudinal direction of the frame forward [math]\displaystyle{ (\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0) }[/math], and the springs deform with opposite signs (one stretches and the other shortens, Figure D1.3a); if it is oriented in the transverse direction of the wagon, [math]\displaystyle{ \Ps }[/math] initially moves in the transverse direction of the frame forward [math]\displaystyle{ (\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0) }[/math], and the change of the springs length is the same (Figure D1.3b); if it is oriented in any other direction, the initial velocity of P has two components ([math]\displaystyle{ \left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0 }[/math],[math]\displaystyle{ \left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0 }[/math], Figure D1.4c). To summarize: the result of this experiment is [math]\displaystyle{ \vel{P,t}{R} \neq 0 }[/math], and it is independent from the position and the time instant, but not from the orientation. For this experiment, the space of that reference frame is homogeneous but not isotropic, and time is uniform.
If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.
- If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the [math]\displaystyle{ \Ps }[/math] motion from rest relative to the platform depends on the initial position and the orientation of the frame: if [math]\displaystyle{ \Ps }[/math] is initially at the centre of the platform, it does not move independently of the orientation of the frame ([math]\displaystyle{ \vel{P,t}{R} = \vec{0} }[/math], Figure D1.4a). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, [math]\displaystyle{ \Ps }[/math] initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed ([math]\displaystyle{ (\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0) }[/math], Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, [math]\displaystyle{ \Ps }[/math] initially moves in the transverse direction of the frame outwards, but the two springs have the same length ([math]\displaystyle{ (\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0) }[/math], Figure D1.4b); If the frame is placed in any other orientation relative to the platform, the initial velocity of [math]\displaystyle{ \Ps }[/math] has two components, and the springs have different lengths ([math]\displaystyle{ (\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0) }[/math], Figure D1.4c). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, the space of the reference frame is neither homogeneous nor isotropic, but time is uniform.
In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (Figure D1.5).
A reference frame where time is uniform and space is homogeneous and isotropic is called a Galilean or inertial reference frame. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called inertia forces – that come from the reference itself.
But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the Principia Mathematica, before stating his laws, Newton postulates the existence of a Galilean reference frame.
Newton's first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in sections D1.4 and D1.5.
D1.2 Galileo’s Principle of Relativity
A principle of relativity establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.
Newtonian mechanics is also based on a principle of relativity: Galileo's Principle of Relativity. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:
[math]\displaystyle{ \text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}), }[/math]
where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.
D1.3 Newton’s Principle of Determinacy
In Newton’s Principia Mathematica, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle [math]\displaystyle{ \Ps }[/math] of the system at time [math]\displaystyle{ t_0 }[/math] depends exclusively on the mechanical state (positions and velocities) of the system at time [math]\displaystyle{ t_0 }[/math]. Mechanical systems, therefore, do not have memory.
Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:
[math]\displaystyle{ \text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right] }[/math]
Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.
D1.4 Newton’s first law (inertia law)
The law of inertia is the solution to the simplest dynamic problem we can imagine: that of the free particle [math]\displaystyle{ \Pll }[/math] (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed:
[math]\displaystyle{ \text{Newton's first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0} }[/math]
In fact, what constitutes a law (and therefore cannot be proved) is that [math]\displaystyle{ \accs{$\Pll,t_0$}{Gal}=0 }[/math]. Indeed:
- If at time [math]\displaystyle{ \ts_0 }[/math] we observe [math]\displaystyle{ \Pll }[/math] from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, [math]\displaystyle{ \vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0} }[/math] (Figure D1.6a).
- If at time [math]\displaystyle{ \ts_0 }[/math] the speed is not zero ([math]\displaystyle{ \vgal{\Pll,\ts_0}\neq\vec{0} }[/math]), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, [math]\displaystyle{ \vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0 }[/math] (Figure D1.6b).
In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that [math]\displaystyle{ \accs{$\Pll,t_0$}{Gal}\neq0 }[/math], and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.
Some people claim that Newton's first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where [math]\displaystyle{ \acc{$\Pll$}{R} = \vec{0} }[/math] (R=Gal). The references where [math]\displaystyle{ \acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0} }[/math] are not Galilean (R=NGal).
If [math]\displaystyle{ \acc{$\Pll$}{R}=\vec{0} }[/math] is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.
Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).
D1.5 Newton’s second law (fundamental law of dynamics)
Newton’s second law formulates the dynamics of the material particle P that interacts with other material particles Q:
[math]\displaystyle{ \sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P}) }[/math],
where [math]\displaystyle{ \ms_\Ps }[/math] is the P mass, and [math]\displaystyle{ \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}} }[/math] are the forces exerted on P by the Q particles.
The [math]\displaystyle{ \ms_\Ps }[/math] parameter appears to be an intrinsic characteristic of P, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.
In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles P and Q. Thus, the sum on the left vanishes, and:
- Due to the space isotropy in Galilean reference frames, the acceleration of P generated by Q must necessarily have the direction [math]\displaystyle{ \QPvec }[/math]. It is an acceleration of attraction (approach) or repulsion (separation).
- Consequently, the [math]\displaystyle{ \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}} }[/math] force that Q exerts on P will also have the [math]\displaystyle{ \QPvec }[/math] direction.
- If we eliminate the Q particle, the parameter [math]\displaystyle{ \ms_\Ps }[/math] becomes irrelevant: [math]\displaystyle{ \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0} }[/math] regardless of its mass. Mass, then, only manifests itself in interaction.
the Let us now go back to the general case of many particles interacting with P. Newton's second law contains a principle of superposition: the resultant force acting on P is the sum of those that each of the other particles would separately exert on P: the simultaneous existence of several particles does not alter the interactions between them (Figure D1.7).
Newton's second law can be used to solve both types of dynamic problems described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.
To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in unit D2.
Newton's second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.
D1.6 Newton’s third law (action-reaction principle)
The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles ([math]\displaystyle{ \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}} }[/math]) must have the direction [math]\displaystyle{ \QPvec }[/math] defined by the particles. Newton's third law ensures that they must be attractive or repulsive, and must have the same value (Figure D1.8).
This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton's most important law because it introduces symmetry into the description of interactions: each interaction is described by a single magnitude. "Force" is not something possessed by a particle, but is associated with pairs of particles.
From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the Principia. Given two interacting particles P and Q, since the mutual force they exert on each other has the same value, their mass ratio ([math]\displaystyle{ \mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps }[/math]) coincides with their acceleration ratio (Figure D1.9). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an inertial interpretation of mass.
If we choose a mass standard (a particular value of mass for a particular particle, for example [math]\displaystyle{ \ms_\textrm{Q}= }[/math]1Kg), the mass of every other particle is determined (Figure D1.10). Newton postulates that this mass is constant and intrinsic to each particle.
D1.7 Particle dynamics in non Galilean reference frames
As stated in section D1.1, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:
Dynamics in Galilean reference frames: [math]\displaystyle{ \sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P}) }[/math]
Dynamics in non Galilean reference frames: [math]\displaystyle{ \sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\Fcal{-}{NGal\rightarrow\Ps}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P}) }[/math]
Subtracting the first equation from the second:
[math]\displaystyle{ \Fcal{-}{NGal\rightarrow\Ps}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P}) }[/math]
If we do a composition of movements to relate [math]\displaystyle{ \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P}) }[/math] and [math]\displaystyle{ \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P}) }[/math]:
[math]\displaystyle{ \left.\begin{array}{l} \mathrm{AB}=\mathrm{Gal} \\ \mathrm{REL}=\mathrm{NGal} \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P}) }[/math]
The inertia force may be decomposed into two forces: the inertia transportation force and the Coriolis inertia force:
[math]\displaystyle{ \Fcal{-}{NGal\rightarrow\Ps}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \Fcal{-tr}{NGal\rightarrow\Ps}+\Fcal{-Cor}{NGal\rightarrow\Ps} }[/math].
Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference frames are distinguishable in Newtonian dynamics!).
It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.
In static situations (absence of movement of P relative to NGal, [math]\displaystyle{ \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0} }[/math]) or when the reference NGal has a translational motion relative to a Galilean reference frame ([math]\displaystyle{ \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0} }[/math]), the Coriolis inertia force is zero.
✏️ Example D1-7.1
- Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively.
- If Q studies the dynamics of P from the ground reference, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.
- If Q studies the dynamics of P from the ground reference, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.
- If P studies the dynamics of Q from the rotating platform reference frame (which has the centre O fixed to the ground and rotates with constant angular velocity [math]\displaystyle{ \Omega_0 }[/math]), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:
- Inertia transportation force [math]\displaystyle{ \Fcal{-tr}{Plat\rightarrow\Qs} }[/math]: as the Q transportation motion is circular with centre O, radius R ([math]\displaystyle{ =\mid \OQvec \mid }[/math]) and associated angular velocity [math]\displaystyle{ \Omega_0 }[/math], the transportation force will be radial and outwards (centrifugal) with value [math]\displaystyle{ \ms\Rs\Omega_0^2 }[/math].
- Coriolis inertia force [math]\displaystyle{ \Fcal{-Cor}{Plat\rightarrow\Qs} }[/math]: it is obtained from the Coriolis acceleration:
[math]\displaystyle{ \Fcal{-Cor}{Plat\rightarrow\Qs}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q}) }[/math]
As the transportation acceleration is orthogonal to [math]\displaystyle{ \OQvec }[/math] and [math]\displaystyle{ \overline{\boldsymbol{\Omega}}_0 }[/math] is vertical, the Coriolis force is radial centripetal (towards O), with value [math]\displaystyle{ 2\ms\Rs\Omega_0^2 }[/math].
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