basic_tools:variational_calculus

This shows you the differences between two versions of the page.

Both sides previous revision Previous revision | |||

basic_tools:variational_calculus [2021/04/17 12:41] cleonis [Why it is interesting?] |
basic_tools:variational_calculus [2021/04/17 19:03] (current) cleonis Added assertion that different approaches will converge to the same differential equation |
||
---|---|---|---|

Line 6: | Line 6: | ||

- | | + | **Variational calculus of the catenary** |

- | ===== Variational calculus of the catenary ===== | + | |

'Catenary' is the name of the curve that represents the shape of a hanging chain. The catenary problem is a problem in statics. Each point of the hanging chain is motionless, as if //all// points of the chain are anchored. | 'Catenary' is the name of the curve that represents the shape of a hanging chain. The catenary problem is a problem in statics. Each point of the hanging chain is motionless, as if //all// points of the chain are anchored. | ||

Line 13: | Line 12: | ||

At the anchor point the tension in the chain can be calculated as follows: the weight tugging at that point is the total weight of the chain. That total weight is tugging in vertical direction. Then the horizontal component of the chain tension is given by the local angle of the chain. | At the anchor point the tension in the chain can be calculated as follows: the weight tugging at that point is the total weight of the chain. That total weight is tugging in vertical direction. Then the horizontal component of the chain tension is given by the local angle of the chain. | ||

- | Given that the chain is motionless this evaluation can be repeated everywhere along the length of the chain, from the top anchor point to the lowest point, which is in the middle. This means that the state of static equilibrium of the catenary can be expressed in the form of a [[https://en.wikipedia.org/wiki/Catenary#Analysis|differential equation]]. | + | Given that the chain is motionless this evaluation can be repeated everywhere along the length of the chain, from the top anchor point to the lowest point, which is in the middle. This means that the state of static equilibrium of the catenary can be expressed in the form of a [[https://en.wikipedia.org/wiki/Catenary#Analysis|differential equation]]. I will refer to this way of obtaining a differential equation as 'the direct approach'. |

This raises the question: is it fortuitous that the problem of finding the shape of a catenary can be stated as a //differential equation//? | This raises the question: is it fortuitous that the problem of finding the shape of a catenary can be stated as a //differential equation//? | ||

Line 20: | Line 19: | ||

- | ==== Brachistochrone problem ==== | + | **Brachistochrone problem** |

When Johann Bernoulli had presented the Brachistochrone problem to the mathematicians of the time Jacob Bernoulli was among the few who was able to find the solution independently. The treatment by Jacob Bernoulli is in the Acta Eruditorum, May 1697, pp. 211-217 | When Johann Bernoulli had presented the Brachistochrone problem to the mathematicians of the time Jacob Bernoulli was among the few who was able to find the solution independently. The treatment by Jacob Bernoulli is in the Acta Eruditorum, May 1697, pp. 211-217 | ||

Line 36: | Line 35: | ||

If the solution is an extremum for the entire curve then it is also an extremum for any sub-section of the curve, down to infinitisimally short subsections. | If the solution is an extremum for the entire curve then it is also an extremum for any sub-section of the curve, down to infinitisimally short subsections. | ||

- | It follows that if a problem can be stated in variational form, then if a solution exists then there exists a way to restate the condition in the form of a differential equation. | + | It follows that if a problem can be stated in variational form, then if a solution exists there exists a way to restate the condition in the form of a differential equation. |

In the case of the catenary the static equilibrium can be cast in terms of minimizing potential energy. The middle of the chain tends to pull the sides inward. When the sides are pulled inward they are also raised, which increases the potential energy of the sides. So the middle can pull the sides inward only so much. The catenary is an extremum of the global potential energy of the chain. | In the case of the catenary the static equilibrium can be cast in terms of minimizing potential energy. The middle of the chain tends to pull the sides inward. When the sides are pulled inward they are also raised, which increases the potential energy of the sides. So the middle can pull the sides inward only so much. The catenary is an extremum of the global potential energy of the chain. | ||

- | The same reasoning, mirrored, applies in the case of a catenary arch. In the case of a catenary arch the extremum is a maximum of potential energy. | + | The same reasoning, mirrored, applies in the case of a catenary //arch//. In the case of a catenary arch the extremum is a maximum of potential energy. |

For the catenary the variational problem then is: find the curve such that the derivative of the total potential energy with respect to variation is zero. Notice that since the evaluation looks exclusively at the //derivative// it is not known whether the extremum is a minimum or a maximum. This does not present any problem; the catenary and the catenary arch have the same shape; the shape is the solution to the problem. Whether the extremum condition is a minimum condition or a maximum condition is immaterial. | For the catenary the variational problem then is: find the curve such that the derivative of the total potential energy with respect to variation is zero. Notice that since the evaluation looks exclusively at the //derivative// it is not known whether the extremum is a minimum or a maximum. This does not present any problem; the catenary and the catenary arch have the same shape; the shape is the solution to the problem. Whether the extremum condition is a minimum condition or a maximum condition is immaterial. | ||

+ | The two approaches, direct approach and variational approach, converge; the two approaches arrive at the same differential equation. | ||

- | ==== Euler-Lagrange equation ==== | + | **Euler-Lagrange equation** |

The strategy of finding a solution to a problem in variational calculus is to find the way to restate the problem in terms of differential calculus. | The strategy of finding a solution to a problem in variational calculus is to find the way to restate the problem in terms of differential calculus. |

basic_tools/variational_calculus.txt · Last modified: 2021/04/17 19:03 by cleonis

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International