The Lagrangian and Calculus of Uariations#
-Let’s begin with motivation of the use of calculus of variations in physics by Fermat’s Principle.
+Let’s begin with motivation of the use of calculus of variations in physics by Fermat’s Principle.
Fermat’s Principle - Path of Least Time#
+Fermat’s Principle - Path of Least Time#
Fermat’s principle is a successful explanation of Snell’s Law. The principle states that light wishes to travel the least time and to do so when changing medium, light wishes to get out of the slow medium as fast as possible; this is done by traveling at an angle where the path length in the slower medium is the shortest.
The length between two points in a plane follows the equation,
Fermat’s Principle - Path of Least Time
-Functional Derivatives#
+Functional Derivatives#
A function is formally defined as taking some value input that lives in the one space to another value that lives in possibly another.
\[
@@ -574,7 +575,7 @@ Functional Derivatives
-Lagrangian and the Principle of Least Action#
+Lagrangian and the Principle of Least Action#
Now, the Lagrangian is a function that is simply the difference between work and potential. Intuitively though not commonly phrased, the Lagrangian amount of activity relative to its potential.
\[
@@ -611,7 +612,6 @@ Lagrangian and the Principle of Least Action\(x(t)\) that gives an extreme (or stationary) action. Intuitively this extrema should be the minimum (this is not always the case) thus this relation is called the principle of least action.
-
Functional Derivatives
-Lagrangian and the Principle of Least Action#
+Lagrangian and the Principle of Least Action#
Now, the Lagrangian is a function that is simply the difference between work and potential. Intuitively though not commonly phrased, the Lagrangian amount of activity relative to its potential.
\[
@@ -611,7 +612,6 @@ Lagrangian and the Principle of Least Action\(x(t)\) that gives an extreme (or stationary) action. Intuitively this extrema should be the minimum (this is not always the case) thus this relation is called the principle of least action.
-
Lagrangian and the Principle of Least Action#
+Lagrangian and the Principle of Least Action#
Now, the Lagrangian is a function that is simply the difference between work and potential. Intuitively though not commonly phrased, the Lagrangian amount of activity relative to its potential.