Editing Classical Mechanics
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 31: | Line 31: | ||
In two dimensions, rather than infinite, the minimum of a function can be described by an equivalent condition to the derivative being 0. Let <math> F:\mathbb{R}^2\rightarrow \mathbb{R} </math>. Typically we would check the condition <math> \frac{\partial F(x_0,y_0)}{\partial x}=\frac{\partial F(x_0,y_0)}{\partial y}=0 </math> at some point <math> (x_0,y_0)\in \mathbb{R}^2 </math>. Rather than differentiating, we can analyze the finite difference treating the input as a vector: <math> F(\mathbf{x}+\mathbf{h})-F(\mathbf{x}) = G(\mathbf{h})</math> and look at the linear part of <math> G </math>. If <math> F </math> was already linear, then computing its derivative comes simply: <math> F(x,y)=ax+by+c\rightarrow G(h_1, h_2)=ah_1+bh_2 </math>. Note the linear dependence on <math> \mathbf{h} </math>, which will remain even when <math> F </math> has higher order terms: <math> G=ah_1+bh_2+ch_1^2+dh_1h_2+\cdots </math>. The functions in finite dimensions we are used to have derivatives, so their derivatives can be described via the linear part of <math> G(\mathbf{h})=L(\mathbf{h})+R(\mathbf{h}), \, L(\mathbf{h}+\mathbf{h}')=L(\mathbf{h})+L(\mathbf{h}') </math>. In infinite dimensions, we may not always have explicit methods of differentiating, but we can look for the linear part of the difference at shifted inputs. We also have to be sure that the entire linear part is in <math> L </math>, so this puts a condition on <math> R </math>. | In two dimensions, rather than infinite, the minimum of a function can be described by an equivalent condition to the derivative being 0. Let <math> F:\mathbb{R}^2\rightarrow \mathbb{R} </math>. Typically we would check the condition <math> \frac{\partial F(x_0,y_0)}{\partial x}=\frac{\partial F(x_0,y_0)}{\partial y}=0 </math> at some point <math> (x_0,y_0)\in \mathbb{R}^2 </math>. Rather than differentiating, we can analyze the finite difference treating the input as a vector: <math> F(\mathbf{x}+\mathbf{h})-F(\mathbf{x}) = G(\mathbf{h})</math> and look at the linear part of <math> G </math>. If <math> F </math> was already linear, then computing its derivative comes simply: <math> F(x,y)=ax+by+c\rightarrow G(h_1, h_2)=ah_1+bh_2 </math>. Note the linear dependence on <math> \mathbf{h} </math>, which will remain even when <math> F </math> has higher order terms: <math> G=ah_1+bh_2+ch_1^2+dh_1h_2+\cdots </math>. The functions in finite dimensions we are used to have derivatives, so their derivatives can be described via the linear part of <math> G(\mathbf{h})=L(\mathbf{h})+R(\mathbf{h}), \, L(\mathbf{h}+\mathbf{h}')=L(\mathbf{h})+L(\mathbf{h}') </math>. In infinite dimensions, we may not always have explicit methods of differentiating, but we can look for the linear part of the difference at shifted inputs. We also have to be sure that the entire linear part is in <math> L </math>, so this puts a condition on <math> R </math>. | ||
=== Infinite Dimensional Techniques === | === Infinite Dimensional Techniques === |