Science & Technology
Karma (कर्म): The Concept of Motion and Equations of motion
Karma (Motion):
Vaiśeṣika (वैशेषिक) uses the term karma for motion in general. It includes five types of karma:
- upward motion (utkṣepaṇa, उत्क्षेपण)
- downward motion (avakṣepaṇa, अवक्षेपण)
- shearing motion (ākuñcana, आकुञ्चन)
- tensile motion (prasāraṇa, प्रसारण)
- rectilinear motion (gamana गमन)
- उत्क्षेपणं शरीरावयवेषु तत्सम्बध्देषु च यदूर्ध्वभाग्भिः प्रदेशैः संयोगकारणमधोभाग्भिश्च प्रदेशैः विभागकारणं कर्मोत्पद्यते, गुरुत्वप्रयत्नसंयोगेभ्यस्तत् उत्क्षेपणम् |
- तद्विपरीतसंयोगविभागकरणं कर्मापक्षेपणम् |
- ऋजुनो द्रवस्याग्रावयवानां तद्देशैर्विभागः संयोगश्च मूलप्रदेशैर्येन कर्मणावयवी कुटिलं संजायते तदाकुञ्चनम् |
- तद्विपर्ययेण संयोगविभागोत्पत्तौ येन कर्मणावयवी ऋजुः सम्पद्यते तत्प्रसारणम् |
- यदनियतदिक्-प्रदेशसंयोगविभागकारणं तद्गमनमिति | एतत्पञ्चविधमपि कर्म शरीरावयवेषु तत्सम्बन्धेषु च सत्प्रत्ययमसत्प्रत्ययं च यदन्यत्तदप्रत्ययमेव तेष्वन्येषु च तद्गमनमिति |
– Praśastapādabhāṣya (प्रशस्तपादभाष्य)
With reference to the above types of motion, ‘gamana karma’ seems to be the general term, which may include the other four types as special cases. The Vaiśeṣika term gamana, i.e., the motion in unspecified direction (aniyata dik, अनियत दिक्) can be treated as a general term for rectilinear motion of physics. When a designation (upādhi उपाधि) is imposed on the direction, viz., motion of the body in upward direction, motion of the body in downward direction, motion of the constitutive elementary parts of the body within it along an axis, or about it, the same gamana is specifically referred to as utkṣepaṇa, avakṣepaṇa, ākuñcana or prasāraṇa respectively.
The Causes of Karma:
The ‘karma’ motion is observed under the action of ‘vega (वेग)’ force on a body. Vaiśeṣika describes the causes of the karma or the motion as well as the measurement of the gamana or the movement. The commentary on Vaiśeṣika provides an in-depth study of the units. It elucidates various types of forces like the mechanical force (Yāntrika bala, यान्त्रिक बल), gravitational force (Gurutva, गुरुत्व), elasticity (Pratyāsthatā (प्रत्यास्थता) or Sthitisthāpakatā (स्थितिस्थापकता)) and other invisible forces like the magnetic forces generated through the gravity.
The causes for the first and second type of motions are gravity: force of attraction of the Earth on the body (gurutva) and mechanical action (prayatna, प्रयत्न). The third and fourth are due to elastic force acting between the molecules of the body and mechanical actions, and the fifth is due to mechanical action only.
According to Vaiśeṣika, Vega (force) is the cause for Karma (motion) and is also proportional to it. Now if the same force is applied on two bodies, the second being of double the mass of the first, the resultant velocity for the heavier body will be just half to that of the first for the same time interval.
Velocity (Gati, गति):
Motion of a body depends on the mass (pārthivamānaṁ, पार्थिवमानं) and velocity (gati) of the body. Bodies move through space. The properties of the space are conjunction, disjunction, number, and unit. All those put together will give us the position of the body relative to the point of observation.
If a body is moving along a straight line in space, and if at equal intervals of time, it is describing equal distances, the body will be said to be moving with uniform velocity (gati) or constant velocity (samagati, समगति). Velocity is defined by the rate of change of distance described by the body. The distance is measured by conjunction and disjunction with the marks placed in the direction, and the time, by a clock.
If a body describes distance ‘s’ with uniform velocity throughout the time ‘t’, the velocity = s/t. Acceleration (gati caya, गति चय) is defined as the rate of change of velocity.
If ‘ u ‘ is the initial velocity and ‘v’ is the final velocity after an elapse of time, then the acceleration is given by:
a = v- u /t
Equations of Motion:
According to physics, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables of spatial coordinates and time but may include momentum components.
There are two descriptions of motion: Kinematics and Dynamics. Kinematics deals with motion where the force is not considered. Dynamics considers force and energy.
There are three equations of motion that use kinematic variables such as displacement (s), velocity (initial (u) and final (v)), time (t) and acceleration (a).
First Equation of Motion: v = u + at
Second Equation of Motion: s = ut + 1/2at2
Third Equation of Motion: v2 = u2 + 2as
These equations of motion can be easily derived by the application of the rules as stated in Līlāvatī (लीलावती) by Bhāskarācārya (भास्कराचार्य), ~12th c. CE.
The First Equation of Motion:
If a body is under uniform acceleration (gati caya) ‘a’, and ‘u’ is the initial velocity when time is zero, then at the end of each second from the start, the velocity will be u, u + a, u + 2a,… and so on.
If the final velocity is ‘v’ and time elapsed be ‘t’ then value of ‘v’ will be given by the formula of Bhāskarācārya. व्यैकपदघनचयो मुखयुक् स्यात् अन्त्यधनम्
According to ancient mathematics, the first term is ‘mukha (मुख)’ which is ‘u’, the number of terms (pada, पद) is (t + 1), common difference of the above series (caya) is ‘a’ and the last term (antyadhanam, अन्त्यधनम्) is ‘v’.
Therefore,
the last value or term = ((Total terms – 1) x common difference) + the first term
or v = at+ u or v= u +at
The Second Equation of Motion:
This equation has been given in a lucid manner by the following sūtra in Līlāvatī.
मुखयुक् दलितं तत् (अन्त्यधनं) मध्यधनम् |
This equation is to determine the distance described by the body in given time ‘t’, if the body is under uniform acceleration, the distance described by the body will be equal to that described by the average velocity in time ‘t’. The average velocity is given by Bhāskarācārya’s formula as:
The average velocity (madhya, मध्य) = [The first term (mukha) + last term (antya)] / 2
= (u + u + at)/2 = u + 1/2 at
Therefore, the distance ‘s’ described in time ‘t’
= (average velocity) x (time)
= (u + 1/2 at) t = u t + 1/2 at2
or s = u t + 1/2 at
The Third Equation of Motion:
The third equation is obtained by eliminating ‘t ‘ from the first two equations.
As from the first equation, we have: t = v- u /a
Substituting the value of ‘t’ in the second equation, we have:
s = u (v- u /a) + 1/2 a (v- u /a)2
or v2 = u 2 + 2as
In this way, we can derive the three equations of motion.
All these equations are valid only for pure rectilinear motion of the bodies.
Vega Saṁskāra (वेग संस्कार): Mechanical Force and Newton's Laws of Motion
Saṁskāra (संस्कार):
Praśastapādabhāṣya (प्रशस्तपादभाष्य) enumerates the three forms of force (saṁskāra), viz. mechanical (vega), emotional (bhāvanātmaka, भावनात्मक) and elastic (Sthitisthāpakatā, स्थितिस्थापकता or Pratyāsthatā, प्रत्यास्थता). Bhāvanā is the property of soul (ātman, आत्मन्).
Elastic force has been considered in the Guṇa (गुण) section. Here we will consider mechanical force (vega) only.
The definition of mechanical force as given in Praśastapādabhāṣya is as follows:
१. वेगः निमित्तविशेषात् कर्मणो जायते |
२. वेगः अपेक्षात् कर्मणो जायते नियत-दिक्-क्रिया-प्रबन्ध-हेतुः |
३. वेगः संयोगविषेशविरोधी, क्वचित् कारणगुणपूर्वक्रमेणोत्पद्यते |
In all the five fundamental physical quantities, i.e, solid, liquid, gas, energy, and mind of Vaiśeṣika (वैशेषिक) the mechanical force is inferred as a specific cause of and as proportional to generated motion (karma, कर्म). The force is directly proportional to motion (karma), caused by it to act in a given direction (as demanded by karma). Like temperature (sparśa, स्पर्श), force also counteracts material conjunction, and sometimes one vega produces the other in series.
If the above definition is broken into three parts, we have the following characteristics of force:
(i) Force is the cause of motion.
(ii) The motion (of an object) is proportional to the magnitude of force and acts in the same direction as the force, and
(iii) Force counteracts (virodha, विरोध) material conjunction (saṁyoga, संयोग) and sometimes one force (vega) produces the other in tandem.
Laws of motion:
First Law of Motion:
The meaning of the word ‘vega’ can be found in Sanskrit grammar. Accordingly, vega is equal to ‘vejana (वेजन)’ and ‘vejayati (वेजयति)’ meaning that which causes a body to move. Thus, vega means the force. Force generates motion, i.e., a change in motion occurs till the force is active.
Newton’s first law of motion:
This law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.
Second Law of Motion:
This law provides a way for measuring force. So long as vega saṁskāra acts, increase in motion (momentum) is observed. In order to define its magnitude, consider the action in unit time, i.e. the change in motion (momentum) is to be considered in unit
time. This consideration will be mathematically equivalent to the imposed force being directly proportional to the rate of change of momentum with the change taking place in the direction of the force.
Newton’s second law of motion:
States that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.
Third Law of Motion:
Kārya (कार्य) or reaction is always in opposition to Karma (motion in action). According to the third law of motion, the mutual actions of two bodies are always equal and act in opposite directions.
It states that force should be exerted in the direction opposite to which work has to be obtained. The examples are rowing a boat, walking, climbing a tree, birds flying in the air, etc. In all these cases, it is observed that to have motion in a desired direction, force has to be applied opposite to it. In case of rowing a boat, water is pressed by the oars in a direction opposite to that of motion and while walking, the feet have to be pressed to the ground opposite to the direction of motion.
From the above examples, it is obvious that the reaction of an action is the only cause for the desired motion. In every case stated above, we find that if a body exerts on the other body, it counteracts upon the first. This reaction (vega) imparts motion to the first body in a direction opposite to that in which it has previously acted upon the second body.
Newton’s third law of motion:
States that all forces between two objects exist in equal magnitude and opposite direction.
The statement of the definition of vega (force) as given by Vaiśeṣika shows that the force is a physical quantity which is inferred by the motion and is only a means for the study of motion and not necessarily an actual existing physical quantity. This concept of the force does not rule out the possibility of ‘fictitious forces’, a concept which is introduced in Einstein’s ‘Theory of Relativity’ and quantum mechanics.