P R O G R A M A introductory specialty exam in mechanics for obtaining an educational master’s degree in specialty 113 – “Applied Mathematics” (educational program – “Theoretical and Applied Mechanics”) (educational-professional level)


  1. Speed ​​and particle acceleration. Determination of them in different ways of the problem of motion of the particle, including, in curvilinear coordinates.
  2. The field of velocities and the field of acceleration of an absolutely rigid body. Angular speed and angular acceleration.
  3. Complex motion of a particle. Adding speeds and accelerations.
  4. Theorem on the change in the number of motion of a particle system. Center of the masses of the system and its movement. Theorem on the change of the kinetic moment of a particle system.
  5. Theorem on the change of the kinetic energy of a particle system. Potential forces. The law of conservation of complete mechanical energy.
  6. Non-free system of particles. Elm Virtual migration is also possible. Number of degrees of freedom of the system of particles. The general equation of the dynamics of systems with ideal joints.
  7. Equation of Lagrange in independent generalized coordinates. Lagrange’s function.
  8. Conditions of equilibrium of mechanical systems. Equilibrium conditions for a free solid and a fixed point body.
  9. Stability of equilibrium for Lyapunov. Lagrange-Dirichlet theorem on the stability of the equilibrium of the conservative system.
  10. Small variations of the conservative system with two degrees of freedom. Main coordinates. Major fluctuations.
  11. Dissipative forces, dissipative Relay function. Influence of dissipative forces on small oscillations of a system with one degree of freedom.
  12. Forced oscillations of the system with one degree of freedom, taking into account and without taking into account dissipative forces. Resonance.
  13. Tensor of solid state inertia. The main axis and the main moments of inertia. Kinetic energy and kinetic moment of the solid.
  14. The equation of motion of a rigid body around a fixed point (Euler’s equation).
  15. Canonical Hamilton equations. The first integrals.
  16. The variational principle of Hamilton-Ostrogradsky.
  17. Curvilinear coordinate system. Basic and mutual bases. Covariant and contravariant components of the vector, the connection between them.
  18. Concept of tensor of the 2nd rank. Tensor algebra. Main axes, main values, invariants of the second-rank symmetric tensor.
  19. Covariant derivative component of vectors and tensors. Christopher’s symbols. Covariant form of the recording of the main relations of continuous media mechanics. Physical components of the vector and the tensor.
  20. Gradient, divergence, and rotor of a vector field, their calculation in curvilinear coordinates. The Gauss-Ostrogradsky and Stokes theorems.
  21. Kinematics of a continuous medium. Tensors of deformations and deformation rates. Relaxing elongations and landslides.
  22. Basic equations of continuum mechanics. Stress tensor. Closing relationships.
  23. Ideal liquid. Euler’s Equation. Euler-Bernoulli, Cauchy-Lagrange Integrals.
  24. Helmholtz theorems on vortices.
  25. The construction of a plane hydromechanics problem of an ideal fluid in terms of velocity potential, current function and complex potential.
  26. Complex potentials of homogeneous translational flow, flows from a vortex and a source. Wrap around the circular cylinder. Dalamer paradox.
  27. Newtonian fluid fluids. Navier-Stokes equation. The flow of a liquid liquid between parallel plates.
  28. Boundary layer Prandtl’s equation.
  29. Turbulent movement of a viscous liquid. Reynolds equation.
  30. The propagation of small perturbations (sound waves) in an ideal gas.
  31. Shock waves Adyabata Hugoniot. Thememplan theorem.
  32. Strain-strain state; Diagram of stretching, Hooke’s law. Physical constants of the theory of elasticity.
  33. The equation of the theory of elasticity in displacements and stresses.
  34. Boundary conditions in the theory of elasticity. The principle of Saint-Venan.
  35. Circumcision of cylindrical rod.
  36. The problem of bending a cylindrical rod.

Approved at the meeting of the Academic Council of the Faculty of Mathematics and Informatics, Minutes No. 1 dated 07.02.2017

Dean of the Faculty of Mathematics and Informatics G. M. Zholtkevich

Approved at the meeting of the admissions committee, protocol No. 2 dated 10.04.2017

O. Anoshenko, Executive Secretary of the Admissions Committee