Researches in the Calculus of Variations Principally On the Theory of Discontin
Researches in the Calculus of Variations Principally On the Theory of Discontin
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Hence if an arc of a parabola alone will not satisfy the conditions, we shall be constrained to adopt the solution given by the diagrams of Art. 167. If we ascribe any variation to the figures, the variation of the action to the first order will be zero for the parabolic part and positive for the rectilinear part : 160 PROBLEM OF LEAST ACTION.
the term of the second order in the variation of the action is necessarily positive as in Art. 170. Hence we are certain that we have a minimum ; and thi...s minimum must give us the least action, as there is no other minimum.
The position of C in the diagrams is easily determined ; for the vertical height of C is known ; and thus the latus rectum of the parabola is known; and so we can find the horizontal distance between C and A or B.
179. The investigations of the present Chapter naturally suggest for discussion the following problem : a particle is pro- jected from a given point with a given velocity, and is attracted to a fixed point by a force varying inversely as the square of the distance : determine the path of minimum action to a second fixed point.
What to read after Researches in the Calculus of Variations Principally On the Theory of Discontin?
You can find similar books in the "Read Also" column, or choose other free books by Todhunter, I. (Isaac), 1820-1884 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Hence if an arc of a parabola alone will not satisfy the conditions, we shall be constrained to adopt the solution given by the diagrams of Art. 167. If we ascribe any variation to the figures, the variation of the action to the first order will be zero for the parabolic part and positive for the rectilinear part : 160 PROBLEM OF LEAST ACTION.
the term of the second order in the variation of the action is necessarily positive as in Art. 170. Hence we are certain that we have a minimum ; and thi...s minimum must give us the least action, as there is no other minimum.
The position of C in the diagrams is easily determined ; for the vertical height of C is known ; and thus the latus rectum of the parabola is known; and so we can find the horizontal distance between C and A or B.
179. The investigations of the present Chapter naturally suggest for discussion the following problem : a particle is pro- jected from a given point with a given velocity, and is attracted to a fixed point by a force varying inversely as the square of the distance : determine the path of minimum action to a second fixed point.
What to read after Researches in the Calculus of Variations Principally On the Theory of Discontin?
You can find similar books in the "Read Also" column, or choose other free books by Todhunter, I. (Isaac), 1820-1884 to read onlineMoreLess
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