ABSTRACT
A key point in the application of variational principles to mechanical systems is Hamilton's principle. A system is called conservative if the forces acting on the system can be expressed as the gradient of some scalar function which is called the potential of the force. This chapter considers a simplified model for the vertical take off of a plane in order to find out the optimal thrust distribution so that the plane achieves a specific height in minimum time. It expresses that the total amount of fuel available is constant. The chapter discusses the modeling of transverse vibrations in elastic bars and thin plates using variational principles. Variational formulation of differential equations can be used to obtain approximate solutions for the equations. The basic technique is due to Rayleigh-Ritz, and it was the "precursor" to the current finite element methods that are used for the numerical solution of partial differential equations in various applications.
