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The starting point is the action, denoted 
(calligraphic S), of a physical system. It is defined as the integral of the Lagrangian L between two instants of time t1 and t2 - technically a functional of the N generalized coordinates q = (q1, q2, ... , qN) which define the configurationof the system:
where the dot denotes the time derivative, and t is time.
Mathematically the principle is[19][20][21]
where δ (Greek lowercase delta) means a small change. In words this reads:[18]
The path taken by the system between times t1 and t2 and configurations q1 and q2 is the one for which the action is stationary (no change) to first order.
In applications the statement and definition of action are taken together:[22]
The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i.e. the curve q(t), parameterized by time (see also parametric equation for this concept).
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FAQs on Principle of least action - Classical Mechanics, CSIR-NET Mathematical Sciences - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is the principle of least action in classical mechanics? |  |
Ans. The principle of least action states that the path taken by a physical system between two points in space and time is the one that minimizes the action, which is a mathematical quantity defined as the integral of the Lagrangian over time. This principle is a fundamental principle in classical mechanics and provides a way to determine the equations of motion for a system.
| 2. How is the principle of least action related to the CSIR-NET Mathematical Sciences exam? | |
Ans. The principle of least action is a topic that is included in the syllabus of the CSIR-NET Mathematical Sciences exam. It is an important concept in classical mechanics and understanding it is crucial for solving problems related to the motion of particles and systems. Questions related to the principle of least action may be asked in the exam to test the candidate's understanding of this topic.
| 3. Can you explain the concept of action in classical mechanics? | |
Ans. In classical mechanics, action is a mathematical quantity that is defined as the integral of the Lagrangian over time. It represents the total effect of a system's motion between two points in space and time. The principle of least action states that the actual path taken by a system is the one that minimizes the action. By varying the path and finding the path that minimizes the action, we can determine the equations of motion for the system.
| 4. How is the principle of least action related to the Lagrangian formulation of classical mechanics? | |
Ans. The principle of least action is closely related to the Lagrangian formulation of classical mechanics. In this formulation, the Lagrangian is defined as the difference between the kinetic energy and the potential energy of a system. The equations of motion can then be derived by applying the principle of least action, which states that the path taken by a system is the one that minimizes the action, defined as the integral of the Lagrangian over time.
| 5. Can you provide an example of how the principle of least action is applied in classical mechanics? | |
Ans. Sure! Let's consider a simple example of a particle moving in one dimension under the influence of a gravitational force. The Lagrangian for this system would be the difference between the kinetic energy and the potential energy of the particle. By applying the principle of least action, we can vary the path taken by the particle and find the path that minimizes the action. This will give us the equation of motion for the particle, which is the second law of motion in this case.
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