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Lillensassi
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I'm reading about the Principle of Least Action.
As a prelude to it, we look at the functional
J(x)=[itex]\int f(y(x),y'(x);x) dx[/itex]
where the limits of integration are x_1 and x_2.
We want to find the function y(x) that gives the functional an extremum.
Now, to do this, we write any possible function y(x) as the sum of the sought function y_0(x) and another function η(x) times a constant α, that is:
y(x)=y_0(x)+αη(x)
so our aim is then to minimize the functional J as a functional of α,
J(α)=[itex]\int f(y(α,x),y'(α,x);x) dx[/itex]
and the book says "The condition that the integral have a stationary value, (i.e. that an extremum results) is that J be independent of α in the first order along the path giving the extremum (α=0), or, equivalenty, that
[itex]\frac{\partial J}{\partial α}[/itex]=0 at α=0".
Now I have two questions:
1. Why not just take the derivative of the functional and put it equal to zero, like we've done when maximizing/minimizing things in Calculus class? and
2. If we know how to make α=0, why do we have to go to the trouble of differentiating the function at all? Isn't the conclusion just stating an obvious fact and not bringing any solution to the table?
I hope my reasoning is understandable, I haven't really completely grasped what I'm doing yet...!
p.s. the text is taken from the book Classical Dynamics of Particles and Systems of Marion and Thornton
As a prelude to it, we look at the functional
J(x)=[itex]\int f(y(x),y'(x);x) dx[/itex]
where the limits of integration are x_1 and x_2.
We want to find the function y(x) that gives the functional an extremum.
Now, to do this, we write any possible function y(x) as the sum of the sought function y_0(x) and another function η(x) times a constant α, that is:
y(x)=y_0(x)+αη(x)
so our aim is then to minimize the functional J as a functional of α,
J(α)=[itex]\int f(y(α,x),y'(α,x);x) dx[/itex]
and the book says "The condition that the integral have a stationary value, (i.e. that an extremum results) is that J be independent of α in the first order along the path giving the extremum (α=0), or, equivalenty, that
[itex]\frac{\partial J}{\partial α}[/itex]=0 at α=0".
Now I have two questions:
1. Why not just take the derivative of the functional and put it equal to zero, like we've done when maximizing/minimizing things in Calculus class? and
2. If we know how to make α=0, why do we have to go to the trouble of differentiating the function at all? Isn't the conclusion just stating an obvious fact and not bringing any solution to the table?
I hope my reasoning is understandable, I haven't really completely grasped what I'm doing yet...!
p.s. the text is taken from the book Classical Dynamics of Particles and Systems of Marion and Thornton