# Dynamics and Stability – Exercises Module 6: Euler angles and Variation Analysis

Course week(s) 6. Euler Angles and Variational Calculus

Corresponding to each module, as described in the lectures section, the teachers handed out one or more sample problems. These are worked out examples that illustrate how to work with a certain method.

Next to the given sample problems, several recommended exercises are given. These exercises can be found in the textbook of this course (Analytical Mechanics, by Josef S. Tőrők). More on this textbook can be found in the readings section.

## Recommended exercises:

Solve the exercises in the provided order:

• 3.32
• 3.3 (Question b is asking you to find an expression depending on δy only (not on δy’))
• 3.5 (“Extremal” means “maximum” or “minimum”. Find the Euler-Lagrange equation only, do not solve it)
• 3.9 (Consider the essential boundary conditions y(0)=y(L)=0)
• 3.34 (Find the “missing” boundary conditions and solve the Euler-Lagrange equations: they are simple)
• Extra exercise: Show that if a functional I(y)=∫Fdx is such that the integrand does not depend on x, that is, F=F(y,y’) then the extremals of I are a solution of the first-order differential equation F-(∂F/∂y’)y’=C, where C is a constant. See the answer sheet in case the mathematical symbols in the previous text are not displaying properly in your browser.