General considerations of constrained mechanical systems. Real, possible and virtual displacements. Simultaneous variations: Lagrange's, Jordan's and Gauss's. Lagrange's multipliers. The Lagrange equations of the first kind. Differential variational principles: the D'Alembert-Lagrange principle, Jordan's principle, Gauss's principle. General equation of statics. Generalized coordinates, velocities and accelerations. The D'Alembert-Lagrange principle in terms of generalized coordinates. The Lagrange equations of the second kind for holonomic and nonholonomic systems. The canonical equations of Hamilton. Kane's equations. Quasi-coordinates. The Gibbs'Appell equations. Acceleration energy. The integral variational principle of Hamilton. The form of the Lagrange function for different mechanical systems and corresponding stationarity conditions. Direct methods of variational calculus. Examples start with simple problems and proceed to real engineering applications such as vehicle motion, robotic systems with rigid and flexible segments, application of the Laplace transform method to nonlinear problems.