Resource Lesson
Principal of Least Action
Printer Friendly Version
Let's begin by taking an example of a block sliding across a completely level, frictionless table, as shown in the strobe picture below.
Based on our definition of average speed as total distance divided by total time we can see that the straight line path followed by the dark block images would be the shortest path. Any deviation from this path, for example, the block's checkered position at t = 0.3 seconds, would require a faster speed to still reach the end in the same amount of time.
This observation lays the foundation for our discussion of using energy to determine the shortest path between two points.
To continue our analysis, let's change our focus from the tabletop picture above to a graph of the block's
Position vs. Time
. On this graph, x
_{2}
, will be the optional position through which the block might pass at t = 0.3 seconds.
The method we will use is to determine nature's preferred path is called the Principal of Least Action. The action, S, is a scalar quantity defined by the integral
where L(x, v) is called the Lagrangian. We can restate this algebraically as
where
D
S an incremental value of the action for a time segment
D
t while KE
_{bar}
and PE
_{bar}
represent the average values of the object's kinetic and potential energy during the same interval.
Mathematicians and scientists have discovered that Nature's preferred path will be one having a minimal action - that is, the path that has the smallest value of when all of the
D
S for each time interval are added together. In our example, all of the time intervals,
D
t, represent the same amount of time and the only differences in the paths are the position of x
_{2}
, our deviated position at 0.3 seconds.
So each
D
S equals
Where the expressions
represent the slopes of the line segments between A and x
_{2}
as well as between x
_{2}
and C. Remember that the slope of a position-time graph represents the object's velocity.
The total action between A and C is found by adding
D
S
_{1}
and
D
S
_{2}
.
Note that
D
t has been distributed throughout the terms in the equation.
To find the minimal action, we now need to take the derivative of S with respect to the only value that can change in our equation, x
_{2}
. Since our table's surface is level, PE is constant, that is, never-changing, throughout the 0.5 seconds the block is sliding.
Setting this derivative equal to zero will give us our optimal value for x
_{2}
.
Notice in the last step that we have shown that our original two slopes must be equal to each other to minimize the action. This means that x
_{2}
must lie exactly along the line between A and C, no deviation is allowed.
The fact that the slopes of the segments Ax
_{2}
and x
_{2}
C must be equal to satisfy the Principal of Least Action is equivalent to Newton's Law of Inertia: an object will remain in a state of constant velocity unless acted upon by an outside, unbalanced force.
Related Documents
Lab:
Labs -
A Battering Ram
Labs -
A Photoelectric Effect Analogy
Labs -
Air Track Collisions
Labs -
Ballistic Pendulum
Labs -
Ballistic Pendulum: Muzzle Velocity
Labs -
Bouncing Steel Spheres
Labs -
Collision Pendulum: Muzzle Velocity
Labs -
Conservation of Energy and Vertical Circles
Labs -
Conservation of Momentum in Two-Dimensions
Labs -
Inelastic Collision - Velocity of a Softball
Labs -
Loop-the-Loop
Labs -
Ramps: Sliding vs Rolling
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Rotational Inertia
Labs -
Rube Goldberg Challenge
Labs -
Spring Carts
Labs -
Target Lab: Ball Bearing Rolling Down an Inclined Plane
Labs -
Video Lab: Blowdart Colliding with Cart
Labs -
Video LAB: Circular Motion
Labs -
Video Lab: M&M Collides with Pop Can
Labs -
Video Lab: Marble Collides with Ballistic Pendulum
Resource Lesson:
RL -
APC: Work Notation
RL -
Conservation of Energy and Springs
RL -
Energy Conservation in Simple Pendulums
RL -
Gravitational Energy Wells
RL -
Mechanical Energy
RL -
Momentum and Energy
RL -
Potential Energy Functions
RL -
Rotational Dynamics: Pivoting Rods
RL -
Rotational Kinetic Energy
RL -
Springs and Blocks
RL -
Symmetries in Physics
RL -
Tension Cases: Four Special Situations
RL -
Work
RL -
Work and Energy
Worksheet:
APP -
The Jogger
APP -
The Pepsi Challenge
APP -
The Pet Rock
APP -
The Pool Game
CP -
Conservation of Energy
CP -
Momentum and Energy
CP -
Momentum and Kinetic Energy
CP -
Power Production
CP -
Satellites: Circular and Elliptical
CP -
Work and Energy
NT -
Cliffs
NT -
Elliptical Orbits
NT -
Escape Velocity
NT -
Gravitation #2
NT -
Ramps
NT -
Satellite Positions
WS -
Advanced Properties of Freely Falling Bodies #1
WS -
Advanced Properties of Freely Falling Bodies #2
WS -
Advanced Properties of Freely Falling Bodies #3
WS -
Charged Projectiles in Uniform Electric Fields
WS -
Energy Methods: More Practice with Projectiles
WS -
Energy Methods: Projectiles
WS -
Energy/Work Vocabulary
WS -
Force vs Displacement Graphs
WS -
Introduction to Springs
WS -
Kinematics Along With Work/Energy
WS -
Potential Energy Functions
WS -
Practice: Momentum and Energy #1
WS -
Practice: Momentum and Energy #2
WS -
Practice: Vertical Circular Motion
WS -
Rotational Kinetic Energy
WS -
Static Springs: The Basics
WS -
Work and Energy Practice: An Assortment of Situations
WS -
Work and Energy Practice: Forces at Angles
TB -
Work, Power, Kinetic Energy
PhysicsLAB
Copyright © 1997-2020
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton