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sncum

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[ds][/2]=[dx][/2]+[dy][/2]+[dz][/2] is the distance in 3-dimension b/w 2 points then

how we can start and how we can take functional by this information?

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In summary, the conversation discusses the use of variational calculus to find the shortest distance between two points in three dimensions. The method involves setting the variation of the integral to zero and using the fundamental theorem of variational calculus to obtain the Euler equations. These equations can be solved to yield a straight line. The purpose of this method is to find paths that extremize the value of the integral, which can be applied in various areas of physics. It is commonly used to derive equations of motion and ensure that new theories obey fundamental principles.

- #1

sncum

- 14

- 0

[ds][/2]=[dx][/2]+[dy][/2]+[dz][/2] is the distance in 3-dimension b/w 2 points then

how we can start and how we can take functional by this information?

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- #2

IsometricPion

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- #3

sncum

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- #4

IsometricPion

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If you post what you have done so far I will better know how to help.

- #5

sncum

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- #6

sncum

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Thats why i have not attached my solution

- #7

IsometricPion

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[itex]\int_{(x_0,y_0,z_0)}^{(x_1,y_1,z_1)}\frac{\frac{ \partial{x}}{\partial{s}}\delta( \frac{\partial{x}}{\partial{s}})+\frac{\partial{y}}{\partial{s}}\delta( \frac{\partial{y}}{\partial{s}})+\frac{\partial{z}}{\partial{s}}\delta( \frac{\partial{z}}{\partial{s}})}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}ds=0\Rightarrow\int_{(x_0,y_0,z_0)}^{(x_1,y_1,z_1)}\frac{\frac{ \partial{x}}{\partial{s}}\delta( \frac{\partial{x}}{\partial{s}})}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}ds+\int_{(x_0,y_0,z_0)}^{(x_1,y_1,z_1)} \frac{\frac{ \partial{y}}{\partial{s}}\delta( \frac{\partial{x}}{\partial{s}})}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}ds+\int_{(x_0,y_0,z_0)}^{(x_1,y_1,z_1)} \frac{\frac{ \partial{z}}{\partial{s}}\delta( \frac{\partial{x}}{\partial{s}})}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}ds=0[/itex] taking [itex]U=\frac{\frac{ \partial{q}}{\partial{s}}}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}[/itex] and [itex]dV=\delta( \frac{\partial{q}}{\partial{s}})ds=\frac{\partial{(\delta{}q)}}{\partial{s}}ds=d(\delta{q})[/itex], where q is the appropriate coordinate in each integral, one obtains [itex]\left[\frac{\frac{ \partial{x}}{\partial{s}}\delta{x}+\frac{\partial{y}}{\partial{s}}\delta{y}+\frac{\partial{z}}{ \partial{s}}\delta{z}}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}\right]_{(x_0,y_0,z_0)}^{(x_1,y_1,z_1)}-\int_{(x_0,y_0,z_0)}^{(x_1,y_1,z_1)}\frac{\partial}{\partial{s}}\left[\frac{\frac{ \partial{x}}{\partial{s}}}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}\right]\delta{x}+ \frac{\partial}{\partial{s}}\left[ \frac{\frac{ \partial{y}}{\partial{s}}}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}\right]\delta{y}+\frac{\partial}{\partial{s}}\left[ \frac{\frac{ \partial{z}}{\partial{s}}}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}\right]\delta{z}\,\,ds=0[/itex]. By the fundamental principle of variational calculus the coefficients of the variations must independently vanish. This yields three coupled partial differential equations: [itex]\frac{\partial}{\partial{s}}\left[\frac{\frac{ \partial{x}}{\partial{s}}}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}\right]=0\,\,\,\,\frac{\partial}{\partial{s}}\left[ \frac{\frac{ \partial{y}}{\partial{s}}}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}\right]=0\,\,\,\, \frac{\partial}{\partial{s}}\left[ \frac{\frac{ \partial{z}}{\partial{s}}}{\sqrt{(\frac{ \partial{x}}{\partial{s}})^2+( \frac{\partial y}{\partial{s}})^2+(\frac{\partial{z}}{\partial {s}})^2}}\right]=0[/itex], which are the Euler equations for this problem. Are these steps similar to yours?

Edit: P.S. If you hit the quote button under a post you can see what was typed to produce the post.

Edit: P.S. If you hit the quote button under a post you can see what was typed to produce the post.

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- #8

sncum

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- #9

sncum

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- #10

IsometricPion

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The main purpose of this method is to find paths which extremize the value of the integral. In your problem we wanted to minimize the length of a line. In geometries that are more complex than Euclidean space even this exercise could produce non-intuitive results. Anything that you want to maximize or minimize (or obtain a stationary point for) that you can describe using an integral would be something you could analyze using this technique (assuming the solution is a contiuous function). In Physics variational calculus is used to derive equations of motion (classically, applications of Newton's second law) from equations giving the energy of a system (wikipedia- Hamiltonian Mechanics). These methods apply in all areas of Physics; Newtonian, quantum, relativistic, and just about any other part of Physics I can think of. This formulation of Physics is useful because it usually makes it easy to ensure that new theories/models obey principles such as conservation of energy, conservation of momentum, etc. due to Noether's Theorem.

- #11

sncum

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I am very thankful to you

The Calculus of Variation is a mathematical theory that deals with finding the optimal value of a functional, which is a mathematical expression involving an unknown function. It involves finding the function that minimizes or maximizes the functional, considering different variations of the function.

The Calculus of Variation was developed by mathematicians such as Johann Bernoulli, Leonhard Euler, Joseph-Louis Lagrange, and Adrien-Marie Legendre. However, Euler is often credited as the founder of this field of study.

Euler's Equation is an important equation in the Calculus of Variation, which is used to find the stationary points of a functional. It states that the derivative of the functional with respect to the unknown function must be equal to zero at the optimal solution.

The Calculus of Variation has various applications in fields such as physics, economics, and engineering. It is used to find the optimal path for a moving object, minimize energy consumption, and optimize the shape of structures, among others.

Some key concepts in the Calculus of Variation include the Euler-Lagrange equation, the functional derivative, and the principle of least action. These concepts are used to solve problems involving the optimization of a functional.

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