Calculus of Variations Dependent variables and constraints

In summary, the conversation discusses finding the euler lagrange equations for a function f(x,x',y,y',t) and a constraint g(x,x',y,y',t). The question is whether both equations should have the same constant A in front of the constraint term, or if each equation requires a different constant. The speaker also asks for clarification on the explicit equation of the constraint, which will define the set of generalized coordinates.
  • #1
shedrick94
30
0
If we have a function:

\begin{equation} f(x,x',y,y',t) \end{equation} and we are trying to minimise this subject to a constraint of
\begin{equation} g(x,x',y,y',t) \end{equation}

Would we simply have a set of two euler lagrange equations for each dependent variable, here we have x and y.

Would we insert f(x,x',y,y',t)-Ag(x,x',y,y',t) into both equations, where A is a constant? Or would each equation require a different constant in front of the constraint term g(x,x',y,y',t)??
 
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  • #2
shedrick94 said:
and we are trying to minimise this subject to a constraint of.
g(x,x′,y,y′,t)​
(2)(2)g(x,x′,y,y′,t)\begin{equation} g(x,x',y,y',t) \end{equation}

Would we simply have a set of two euler lagrange equations for each dependent variable, here we have x and y.

pl. give explicit eq of constraint -that will define the set of generalized coordinates.
 

What is the purpose of the Calculus of Variations?

The Calculus of Variations is a mathematical tool used to find the optimal value of a functional, which is a function that takes in other functions as inputs. It is often used in physics and engineering to solve problems related to optimization and control.

What are dependent variables in the context of Calculus of Variations?

Dependent variables in the Calculus of Variations refer to the functions that are being optimized. These functions are dependent on other variables and are often subject to constraints, which can be mathematical equations or physical limitations.

How are constraints incorporated into the Calculus of Variations?

Constraints are incorporated into the Calculus of Variations through the use of Lagrange multipliers. These are parameters that are introduced into the optimization problem to enforce the constraints and find the optimal solution.

What is the relationship between Calculus of Variations and the Euler-Lagrange equation?

The Euler-Lagrange equation is a fundamental equation in the Calculus of Variations that allows us to find the optimal value of a functional. It is derived by setting the derivative of the functional with respect to the dependent variable equal to zero and solving for the dependent variable.

What are some real-world applications of the Calculus of Variations?

The Calculus of Variations has many applications in physics and engineering, such as finding the optimal path for a spacecraft to travel between two points, determining the shape of a bridge that can withstand the most weight, and optimizing the trajectory of a projectile to hit a target with minimal energy. It is also used in economics to model consumer behavior and in medicine to optimize drug dosages.

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