# Variational Calculus : Geodesics w/ Constraints

• lstellyl
In summary, the conversation discusses finding geodesics joining two points on a cylinder and finding extremals of the length functional joining two points. The equations used were Euler-Lagrange generalized equations and the attempt at a solution involved using F and G as defined in the equations and solving for lambda, x, y, and z. However, the solutions obtained did not seem correct and there may be an error in the approach. It is suggested to use G=(x^2+y^2-a^2) and to find lambda in terms of C and a^2. It is also mentioned that there may be easier ways to solve this problem.
lstellyl

## Homework Statement

Consider the cylinder S in R3 defined by the equation $$x^2+y^2=a^2$$

(a). The points $$A=(a,0,0) \: and \: B = (a \cos{\theta}, a \sin{\theta}, b)$$ both lie on S. Find the geodesics joining them.

(b). Find 2 different extremals of the length functional joining $$A=(a,0,0) and C = (a, 0 2 \pi, b)$$. How many extremals join A and C?

## Homework Equations

Euler-Lagrange generalized equations:
$$\frac{\partial}{\partial x_i } \{F+\sum_k \lambda_j (t) G_j \} - \frac{d}{dt} \{ \frac{\partial}{\partial x'_i } \{ F+\sum_k \lambda_j (t) G_j \} \}$$

## The Attempt at a Solution

using $$F= \sqrt{x'^2+y'^2+z'^2} \: and \: G= x^2+y^2$$ and assuming that the geodesic path $$\gamma$$ was parameterized by arc length, meaning that |u'(t)|=1, i was able to get the following equations:

$$\lambda (t) x(t) = x''(t)$$

$$\lambda (t) y(t) = y''(t)$$

$$z'(t) = C$$ where C is a constant.

although for some reason mathematica wouldn't solve this for me... these are all simple harmonic oscillator equations... (hopefully, i say that and actually solve them correctly...)

forming the equations with constants and solving for the boundary conditions, i found the following using Mathematica's Solve function:

$$x(t) = A e^{k t} + D$$
$$y(t) = B e^{k t} + E$$
$$z(t) = \frac{b}{L} \:\: (trivial)$$

where

$$A = - \frac{2 a \sin{\theta/2}^2}{e^{ k L} - 1}$$

$$B = \frac{ a \sin{\theta}}{e^{ k L} - 1}$$

$$D = \frac{ a ( e^{k L} - \cos{\theta})}{e^{ k L} - 1}$$

$$E = B$$

where k = Sqrt[lambda] and L is the length of the curve (since it is parameterized by arc-length)

...

This solution doesn't look right to me, and I don't want to move on to part 2 with this incorrect. I feel like i am doing something implicitly wrong and should maybe be canceling out lambdas or something... because I still have a lambda in those constant values...

any help or direction would be appreciated... it has not been a good day for me and i am having trouble focusing.

You probably actually want to use G=(x^2+y^2-a^2). And yes, they are simple harmonic motion equations, since lambda is a constant. Not a function of t. Your next step after that shouldn't be to resort to Mathematica's Solve function. Those solutions actually look pretty wacky. Your next step should be to try to find lambda in terms of C and a^2. Hint: differentiate x^2+y^2=a^2 twice and use your |u'(t)|=1 condition and the other equations. Do you really have to solve this as a constraint problem? There are easier ways.

## 1. What is variational calculus?

Variational calculus is a branch of mathematics that deals with finding the optimal path or function that minimizes a specific quantity, such as time or energy. It is often used in physics and engineering to solve problems involving constrained optimization.

## 2. What are geodesics?

Geodesics are the shortest paths between two points on a curved surface or in a curved space. These paths are determined by the geometry of the surface or space and are often used to model the motion of objects.

## 3. How is variational calculus used to find geodesics with constraints?

In variational calculus, the geodesic is represented by a functional, which is a mathematical expression that takes in a function and returns a real number. The constraints are then incorporated into the functional as additional conditions that the function must satisfy. The optimal function, and therefore the geodesic, can then be found by minimizing the functional.

## 4. What are some common constraints used in variational calculus for finding geodesics?

Some common constraints used in variational calculus for finding geodesics include fixed endpoints, fixed arc length, and fixed curvature. These constraints can also be combined to model more complex situations, such as a path that must pass through a specific point while maintaining a constant speed.

## 5. What are some applications of variational calculus for finding geodesics with constraints?

Variational calculus and geodesics with constraints have many applications in physics and engineering. They can be used to model the motion of objects in relativity, to optimize the shape of structures for maximum strength, and to find the most efficient flight paths for aircraft. They are also used in computer graphics and animation to create realistic motion and deformation of objects.

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