Newtonian Mechanics 01

— 2. Initial Considerations —

Without being too far away from the truth one can say that modern Fundamental Physics rests on these three conceptions

  1. The concept of field
  2. The theory of relativity
  3. Quantum Physics

The concept of field is essential to all of our future discussion hence we’ll define it right away:

Definition 1 Field is an mathematical structure with a defined value in a given set of points.
Definition 2 A field is said to be a vectoral field when it assumes vector values.
Definition 3 A field is said to be a scalar field when it assumes scalar values.

The field equations that we are going to use always represent linear interactions. Hence one can consider each interaction resulting from a field as being independent of all other interactions being analyzed and the resulting interaction is just the sum of all interactions.

Associated to the concept of field we have the concept of potential energy. This energy arises as a result of the result of the interaction of the particle with the field {\vec{A}} and in general it is proportional to {\displaystyle\int_a^b\vec{A}\cdot d\vec{s}} where {d\vec{s}} represents an infinitesimal displacement.

— 3. Mechanics —

This section will be a very brief and shallow introduction to the results and triumphs of the first modern physical thepry. Nevertheless one hopes that its internal elegance and depth can be glimpsed through.

— 3.1. Basic concepts and preliminary definitions —

All mechanical quantities can be expressed in units that derive from the units of the following quantities:

  • Length {L}.
  • Time {T}.
  • Mass {M}. In classical mechanics the mass of a body its a measure of its resistance to alter its state of movement. This characteristic has the name of inertia.

The units that one uses to express the previous quantities are totally conventional. In the following we’ll use the Internation System of Units:

  • {\left[ L \right] =m}
  • {\left[ T \right] =s}
  • {\left[ M \right] = \mathrm{Kg}}
Definition 4

A frame is a set of axes that represent the degrees of freedom of the system that is being studied and an arbitrary point that serves has its origin.

Definition 5 A frame is said to be inertial when it possesses the following properties:

  • Space is homogeneous (all points are equivalent) and isotropic (there is no special orientation)
  • Time is homogeneous (all time instants are equivalent)
Definition 6 Position is the geometric localization of a particle in a frame.
Definition 7 Trajectory is the geometric place of the sucessive positions of a particle in a frame in a given time interval.
Definition 8 Displacement it’s the difference between the final position and the initial position of a particle. The displacement is represented by the symbol {\Delta \vec{x}}.

We know by everyday experience that bodies move by different displacements during different time intervals. The concept that expresses how a particle position changes in a given time interval it’s called velocity.

Definition 9 Average velocity: vectorial quantity that expresses that rate of change in a particles position for a given time interval:

\displaystyle \vec{v}_m=\dfrac{\Delta \vec{x}}{\Delta t} \ \ \ \ \ (4)

Definition 10 Velocity: Vectorial quantity that expresses a particles velocity in a given time instant:

\displaystyle \vec{v}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta \vec{x}}{\Delta t}=\dfrac{d\vec{x}}{dt} \ \ \ \ \ (5)

Since a particle’s velocity varies in time one can introduce the concept of acceleration.

Definition 11 Average acceleration: Vectorial quantity that expresses the rate of change of velocity for a given time interval.

\displaystyle \vec{a}_m=\dfrac{\Delta \vec{v}}{\Delta t} \ \ \ \ \ (6)

Definition 12 Acceleration: vectorial quantity that expresses the change in velocity in a given time instant.

\displaystyle \vec{a}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta \vec{v}}{\Delta t}=\dfrac{d\vec{v}}{dt} \ \ \ \ \ (7)

Definition 13 Kinetic energy is the energy associated with the movement of a particle and is defined by:

\displaystyle K=\dfrac{1}{2}m\vec{v}\cdot\vec{v}=\dfrac{1}{2}mv^2=\dfrac{1}{2}m\left( \dfrac{d\vec{x}}{dt}\right)^2 \ \ \ \ \ (8)

Definition 14 Linear momentum : vectorial quantity that is associated with the movement of a particle

\displaystyle \vec{p}=m \vec{v}=m \dfrac{d\vec{x}}{dt} \ \ \ \ \ (9)

Definition 15

The mechanical state of a particle is specified by the simultaneous determination (and with infinite precision) of its coordinates and linear momentum.

— 3.2. Newton’s Axioms —

Thus far we have defined the actors in our stage play but we haven’t defined by which rules do they interact. These rules are given by Newton’s Axioms.

Axiom 1 There is an inertial frame where the linear momentum of a free particle (a particle that experiences no interactions) is constant

Our definition 5 of an inertial frame is, strictly speaking, a mathematical definition. As such nothing in it implies that such a frame exists in our world. As such the function of Newton’s first Axiom is to guarantee that in inertial frame exists in our world.

One important detail is that Axiom 5 only posits the existance of one inertial frame, but we can conclude that there exists an infinite number of inertial frames.

Since an inertial frame space is homogenous and isotropic the point one picks as the origin is arbitrary. Hence one can make a translation of the first origin point and consider the final origin point as being the origin of this new frame, which has to be a new inertial frame.

Moreover one can apply a rotation to our firs inertial frame and obtain a new frame (the time the axes have different orientations). Since space is isotropic this orientation can’t change the nature of the frame and we can conclude that the final frame is also an inertial one.

Additionally one can consider and inertial that moves with constant velocity relatively to an inertial frame. Since space is homegeneous this operation can’t change the nature of the the frame in question and once again it still has to be an inertial frame.

Likewise, since time time is homogeneous one can consider a frame that differs of an inertial frame only in the instant of time that was taken as its origin. Since time is homogeneous this second frame also has to be an inertial one.

Finally let us just notice that any finite or infinite composition of these transformations also produces an inertial frame.

Axiom 2

An in inertial frame the change of linear momentum of a particle is caused by the action of a force {\vec{F}}.

{\vec{F}= \dfrac{d\vec{p}}{dt}}.

When the mass of the particle being considered is constant this axiom becomes {\vec{F}=m\vec{a}}.

In what follows the mass of a particle is to be considered constant unless stated otherwise.

Axiom 3

When two particles interact the force {\vec{F}_{12}} (force that object {1} exerts on object {2}) has an equal intensity as {\vec{F}_{21}} but opposite direction{\vec{F}_{12}=-\vec{F}_{21}}

— 3.3. Kinematics and Dynamics —

On this section we’ll introduce very briefly the concepts that describe (kinematics) and explain (dynamics) the movement of a particle.

— 3.3.1. Equations of movement —

From the definitions of velocity and acceleration that we introduced in section 3 it follows that

\displaystyle d\vec{v}= \vec{a}dt \Rightarrow \int_{t_0}^t d\vec{v}= \int_{t_0}^t \vec{a}dt \Rightarrow \vec{v}(t)-\vec{v}(t_0)=\int_{t_0}^t \vec{a}dt \ \ \ \ \ (10)

 

Since the functional relationship between the acceleration and time is unknown the right hand side of the equality can not be calculated.

It also is

\displaystyle d\vec{x}= \vec{v}dt \Rightarrow \int_{t_0}^t d\vec{x}= \int_{t_0}^t \vec{v}dt \Rightarrow \vec{x}(t)-\vec{x}(t_0)=\int_{t_0}^t \vec{v}dt \ \ \ \ \ (11)

 

Since {\vec{v}(t)} is also an unknown function the calculation has to stop.

If we assume that {\vec{a}} is constant in time (uniformly accelerated motion) we can solve equation 10, { \vec{v}=\vec{v}_0+\vec{a}(t-t_0)}. After substituting in equation 11 it follows

\displaystyle \vec{x}(t)=\vec{x}_0+\vec{v}_0(t-t_0)+\frac{1}{2}\vec{a}(t-t_0)^2 \ \ \ \ \ (12)

 

For the special case {\vec{a}=\vec{0}} the motion is said to be uniform.

— 3.3.2. Galileu Transformations —

After axiom 1 we concluded that there is an infinite number of inertial frames. Hence it makes sense to ask how can calculate the coordinates (velocity) of particle in an inertial frame when knowing its coordinates (velocity) in a first inertial frame.

Let {S} and {S'} be two inertial frames whose origins coincide at {t=0}. Moreover {S'} moves with velocity {\vec{v}_0} relative to {S}.

TransformacaoGalileu

By simple vector addition it is {\vec{v}_0 t+\vec{r}'=\vec{r}} which we can write in component form:

{\begin{aligned} x' & = & x-v_{0x}t\\ y' & = & y-v_{0y}t\\ z' & = & z-v_{0z}t \end{aligned}}

Deriving in order to {t} it is

{\begin{aligned} v'_x & = & v_x-v_{0x}\\ v'_y & = & v_y-v_{0y}\\ v'_z & = & v_z-v_{0z} \end{aligned}}

Galileu transformations are equivalent to the affirmation that the form of the equations of mechanics don’t depend on the inertial frame that one chooses to study the motion of a particle.

 

Advertisements