## Abstract

The principles and concepts in describing motion of different bodies on earth can be explained through the science of mechanics. Classical mechanics specifically Newtonian mechanics is the most unified and comprehensive of all the mechanics categories. With this unified principles, many breakthroughs in science were developed.

## Introduction

Motion is always involve in every aspect of our lives; from simple walking to unparalleled velocity of a bullet train. Different ideas and concepts are involved in describing different kinds of motions. Aside from quantum mechanics, classical mechanics comprises the two major fields in physics. This is involved in describing the set of physical laws and mathematical expressions of motions of a particular body.

This paper will tackle the different achievements of classical mechanics from its origin up to recent researches and inventions. This will show how classical mechanics is considered as the unifying category of all the breakthroughs in science.

### Birth of Classical Mechanics

Classical mechanics birth existed as early as the 17th century. Isaac Newton, Galileo Galilei, Joseph Louis Lagrange and William Rowan Hamilton were among the earliest physicist and mathematicians that presented works that build classical mechanics.

### Newtonian Mechanics

In 1687, Sir Isaac Newton (1642-1727) published his elaborated laws of motion in Principia (Mathematical Principles of Natural Philosophy). This work ranks among the most influential scientific books ever published. The book’s publication established the foundation of classical mechanics. Newton’s work on mechanics dealt primarily with particle motion. Another 200 years elapsed before rigid-body dynamics, fluid mechanics and the mechanics of deformable bodies were developed. Each of these areas required new axioms before it assume a usable form.

Newtonian mechanics have even influenced two other branches of mechanics, born at the beginning of the twentieth century: relativistic and quantum mechanics. However, relativistic and quantum mechanics have by no means invalidated the principle of Newtonian mechanics. In the analysis of the motion of bodies encountered in our everyday experience, both theories converge on the equations of Newtonian mechanics.

### Newton’s first law: Law of Inertia

The Law of Inertia states that “A particle at rest (or moving with constant velocity in a straight line) will remain at rest (or continue to move with constant velocity in a straight line) unless acted upon by a force.” (Pytel, 1996) This involved two points:

- An object that is not moving will not move until acted upon by a force;
- An object that is moving will accelerate until a force acts upon it.

Some physicists argue that the second point seems to violate everyday experience. An eraser sliding in a surface slows down and eventually comes to a stop without applying any force to stop it. According to Newton’s first law, there is actually an external force that acts against the moving body. This force is the friction between the eraser and the table. This explains how rolling or sliding of an object stops even without blocking or without applying any external force.

Newton’s first law credited Galileo as it is a restatement of what Galileo had already described. It contradicts Aristotle’s view that all objects have a natural place in the universe. Aristotle’s view is that objects like rocks wanted to be at rest on earth and the stars and smoke on the sky.

Before Newton formulated the first law, several different natural philosophers and scientists already described this particular idea about motion. Among those philosophers are a Chinese philosopher Mo Tzu (3rd century B.C.), Alhazen and Avicenna in the 11th century.

### Newton’s second law

The second law states that “A particle acted on by a force will accelerate in the direction of the force. The magnitude of the acceleration is proportional to the magnitude of the force and is inversely proportional to the mass of the particle.” (Pytel, 1996) Using modern symbolic notation, Newton’s second law can be written as:

- Fnet= m dv/dt=ma where: Fnet= total force
- m= mass
- a= acceleration

This equation is almost applicable to all speeds within the human experience. However, if higher speeds are involved, the approximation of the second law become inaccurate and the theory of the special relativity must be applied.

The second law is closely related to impulse. An impulse occurs when a force F acts at a time range and is stated as òDt Fdt. A much analysis of the second law describes the relation between impulse and change of momentum. The analysis is given by:

- I = Dp= mDv where: I= impulse
- Dp= change in momentum
- Dv= change in velocity

### Newton’s third law

The Newton’s third law states that “For every action, there is an equal and opposite reaction; that is, the forces of interaction between two particles are equal in magnitude and opposite in direction.” (Pytel, 1996)

The third law can be seen in many cases. If you push a big box, the box also pushes you in return. If the force of a body changes the motion of the other, the body will also undergo an equal change, not in velocities but in the motion or momentum of the bodies.

All of the three laws of Newton were verified by experiment and observation for over hundred years. Together with Newton’s law of universal gravitation and the mathematical techniques of calculus, Newton’s laws of motion gave a unified explanation of various physical phenomena. However, the combined concepts of Newton’s laws of motion and universal gravitation could not explain certain phenomena that involve small entities, very high speeds and very strong gravitational fields. With these, Newton’s laws are entirely not valid in certain phenomena such as conduction of electricity and optical properties of substances. These phenomena can be explained through the concepts provided by general relativity and quantum mechanics.

### Inertial reference frames

When applying Newton’s second law, attention must be paid to the coordinate system in which the accelerations are measured. An internal reference frame (also known as Newtonian or Galilean reference frame) is defined to be any rigid coordinate system in which Newton’s laws of particle motion relative to that frame are valid with an acceptable degree of accuracy. In most design applications used in the surface of the earth, an inertial frame can be approximated with sufficient accuracy by attacking the coordinate system to the earth. A definite example is the study of satellites in which a coordinate system is attached to the sun.

### Lagrangian Mechanics

Lagrangian mechanics is one of the abstract and general methods in the advancement of classical mechanics. It is a reformulation of classical mechanics that combines conservation of momentum with conservation of energy. Introduced in 1788 by Joseph Louis Lagrange, an Italian mathematician, Lagrangian mechanics described the trajectory of a system of particles by solving Lagrange’s equation. Solving Lagrange equation gives the path that minimizes the action functional. Action functional is the quantity that is the integral of the Lagrange over time.

### Hamiltonian Mechanics

In 1833, Irish mathematician William Rowan Hamilton introduced the so called Hamiltonian mechanics which is a reformulation of classical mechanics. Hamiltonian mechanics arose from Lagrangian mechanics. It simply differs from Lagrangian approach by its expression of the first-order equations on a 2-n dimensional phase space.

Generally, Hamilton’s equations provide deeper analysis of both the general structure of classical mechanics and its connections to other areas of science. In comparison to Lagrangian equations, Hamiltonian equations are much easier to solve. All in all, it will produce the same results as Lagrangian mechanics and Newton’s laws of motion.

### Branches of Classical Mechanics

Classical mechanics was conventionally divided into three branches: statics, dynamics and kinematics. Statics is the study of objects in equilibrium and its relation to forces. Dynamics on the other hand is the study of motion of objects and its relation to forces. Lastly, kinematics describes the motion of the body regardless of the causes of its movement.

### Related facts on classical mechanics

One of the remarkable achievements of classical mechanics, specifically Newtonian mechanics is the derivation of the Kepler’s law of planetary motion. The idea of motion of a projectile in a uniform gravitational field was first investigated by Galileo Galilei. These breakthroughs initiated the future science of mechanics.

Many other works and discoveries in science adapt classical mechanics as the foundation of their study. Among these are the derivation of Daniel Bernoulli of fundamental frequency of a stretched vibrating string and the derivation of fundamental frequency and harmonics of a hanging chain. In 1739, Leonhard Euler solved the differential equation for a force harmonic oscillator.

## Conclusion

In the complicated world of science, there exist many principles and ideas that are involve in our everyday life. With this many principles, a foundation that are being used for over 500 years is the classical mechanics. It is the fundamental of all the principles of motion and the founding principle of many breakthroughs in science of motion.

## References

- Fitzpatrick, Richard. “Classical Mechanics: An Introductory Course”. Retrieved April 11, 2009 @ http://farside.ph.utexas.edu/teaching/301/301.html
- Poole, Charles. Classical Mechanics (3rd Edition), Addison Wesley
- Pytel, Andrew. “Classical Mechanics: Newtonian Mechanics”. Statics and Dynamics (pp 4-7).Harper- Collins Publishing (1996).