• Mechanics, Semester 1, position 1

-Тo provide students knowledge of the fundamental principles and methods in Analytical Mechanics -Тo enable students to solve practical problems in Analytical Mechanics using acquired knowledge in Analytical Mechanics - to monitoring novelties in science and engineering

-Тo enable students to master terms, methods and principles in Analytical Mechanics -Тo enable students to relate the knowledge from knowledge in other scientific fields with knowledge Analytical Mechanics -Тo apply knowledge from Analytical Mechanics in analysis, synthesis and prediction of solutions and consequences of problems in science - Тo monitoring novelties in science and engineering

Analytic system dynamics. Free and constrained material systems. Constraints and their classification. Real, possible and virtual displacements. Number of degrees of freedom. Virtual work of forces. Ideal constraints. Lagrange’s equations of the first kind. Lagrange’s mechanics and differential approaches. Elements of tensor calculus. State of motion of mechanical system in configuration space. Kinetic energy. Generalized forces. Virtual work principle. Lagrange-D’Alembert’s principle. Lagrange function. Lagrange’s equations of the second kind for holonomic mechanical systems and their structure. First integrals. Lagrange’s equations of the second kind for nonholonomic mechanical systems. Hamiltonian mechanics. Hamilton’s variables. Phase space. Hamilton’s function and its structure. Hamilton’s canonic equations for conservative and nonconservative holonomic mechanical systems. Direct method for finding first integrals of Hamilton’s canonic equations. Poisson bracket. Liouville’s theorem. Whittaker’s equations. Routh’s equations. Variational principles. Elements of variational calculus. Relation between general dynamic equation and variational calculus. Central equation of dynamic. Second form of Hamilton’s principle. Hamilton-Jacobi method of integration of canonic equations. Lagrange principle.

Analytic system dynamics. Free and constrained material systems. Constraints and their classification. Number of degrees of freedom. Virtual work of forces. Ideal constraints. Lagrange’s equations of the first kind. Lagrange’s mechanics and differential approaches. Elements of tensor calculus in analytical mechanics.Kinetic energy. Generalized forces. Virtual work principle. Lagrange-D’Alembert’s principle. Lagrange function. Lagrange’s equations of the second kind for holonomic mechanical systems and their structure. First integrals. Lagrange’s equations of the second kind for nonholonomic mechanical systems. Hamiltonian mechanics. Hamilton’s variables. Hamilton’s function and its structure. Hamilton’s canonic equations for conservative and nonconservative holonomic mechanical systems. Direct method for finding first integrals of Hamilton’s canonic equations. Whittaker’s equations. Routh’s equations. Variational principles. Elements of variational calculus in mechanics. Hamilton-Jacobi method of integration of canonic equations.

Defined by curriculum.

[1] Leko M., Plavšić M., Solved problems from tensor calculus with application in mechanic, Scientific book, Belgrade,1973. [2] Fempl S., Elements of variational calculus, Building book, Belgrade, 1965. [3] Lurje A.I., Analytical mechanics, State publishing house, F.M.literature, Moscow, 1961. [4] Jeremić, O., Written lectures in Analytical mechanics, Faculty of Mechanical Engineering, University of Belgrade, Belgrade, 2020.

Total assigned hours: 75

New material: 20
Elaboration and examples (recapitulation): 10

Auditory exercises: 30
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 0
Review and grading of seminar papers: 0
Review and grading of the project: 0
Test: 10
Test: 0
Final exam: 5

Activity during lectures: 0
Test/test: 60
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Final exam: 40
Requirement for taking the exam (required number of points): 0

Simić S.; Analytical Mechanics, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, 2006.; Vuković J.; Selected topics in Mechanics, Written lectures for PhD studies, Faculty of Mechanical Engineering, University of Belgrade, , Belgrade, 2004.; Gantmaher F. R., Analytical Mechanics, Institute for textbooks, Belgrade, 1963. ; Jeremić O.; Written lectures in Analytical mechanics, Faculty of Mechanical Engineering, University of Belgrade, Belgrade, 2020.; Radovanović V.; Theoretical mechanics, Lagrangian and Hamiltonian mechanics , Physical faculty, University of Belgrade, Belgrade, 2020.