Final answer:
To find the Euler equation in optimisation, identify the knowns and unknowns, select the appropriate equation, take derivatives and substitute them into the Euler equation, and integrate using initial conditions.
Step-by-step explanation:
To find the Euler equation using first order conditions with optimisation, follow this strategy:
- Identify the knowns and unknowns: Look at the problem at hand and determine what variables and constants are provided, and what you need to solve for. For instance, if we are given a utility function ŭ and we take x0-reaction to be 0, we want to find the value for Xreaction.
- Select the appropriate equation: Once the knowns and unknowns are clear, choose the equation that relates these variables. This might be a differential equation whose solution characterizes the optimal path in a dynamic optimization problem.
- Apply the first and second derivatives: Take the first and second derivatives of the function with respect to the relevant variable(s), often time. Then substitute these derivatives into the Euler equation to determine if the obtained function satisfies the equation.
- Integrate with initial conditions: If solving a dynamic problem, integrate the differential equation and apply any given initial conditions to find the constants of integration.
This method can be applied to various fields of study, such as finding the energy (Eo) of an object at rest with mass m using the famous Einstein's equation Eo = mc², or determining the kinetic energy (KErel) related to relativity using KErel =(γ - 1)mc² after calculating gamma (γ).
The first condition of equilibrium in physics states that the net force acting on a stationary object must be zero.