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# MATHEMATICAL METHODS AND MATHEMATICAL PHYSICS FOR APPLICATIONS

 Degree course Civil Engineering Curriculum IDRAULICA Learnings Orientamento unico Academic Year 2014/2015

## Module: MATHEMATICAL PHYSICS FOR APPLICATIONS

 Degree course Civil Engineering Curriculum IDRAULICA Learnings Orientamento unico Academic Year 2014/2015 ECTS 6 Scientific Disciplinary Sector MAT/07 Year First year Time unit Second semester Class hours 48 Educational activity Related and integrative training activities

## Single group

 Professor PASQUALE GIOVINE Objectives The main target of the discipline is the discussion of methodologies useful in the study and mathematical formalization of the various classes of physical phenomena of interest and relevant to interdisciplinary applications of engineering. So - at the end of the course - the student will be able to address and solve numerous problems related to the motion of Lagrangian systems and continuous ones, as well as to their equilibrium; also he will have acquired a basic understanding on partial differential equations and some methods for their resolution. Programme 1. Basic analytical mechanics (1,5 UFCs)Principle of reaction forces - Friction force as an active force - Principle of d'Alembert - Relation and symbolic equation of dynamics - Equations in Lagrangian coordinates of the motion of a holonomic system subject to perfect and bilateral constraints - Generalized forces - Lagrange's equations - Kinetic energy of a holonomic system – Cauchy’s problem for the equations of Lagrange - Kinetic energy theorem – Conservation theorem of mechanical energy for constrained systems - Lagrange's equations in the conservative case - Lagrangian function - Generalized potential - Symmetric and skew operators - Symbols of Kronecker and Levi-Civita: properties and relationships - First integrals of the motion equations: momentum, axial angular momentum and mechanical energy - Lagrangian systems - Kinetic moments – Typical first integrals typical of Lagrangian systems - Acts of motion - Equilibrium of a holonomic system - Principle of stationary potential - Stability of equilibrium - Dirichlet's theorem - Dissipative case - Small oscillations of a holonomic system about a stable equilibrium position2. Matrix operators of vectors (1 UFCs)Matrix operators and Cartesian components - Identity operator - Product of a scalar to a matrix operator - Sum of two operators - The product of two matrix operators - Transpose operator - Trace of an operator - Determinant of an operator: expression of the determinant in the case of n = 3 - Inverse operator - Complementary operator - Some remarkable identities for matrix operators: the case n = 3 - Scalar product between operators - Symmetric and skew operators: dual vector associated with a skew-symmetric operator, symmetric and antisymmetric parts of an operator - Deviatoric and isotropic parts – Rotation operator - Transformations of orthogonal similarity: main invariants of an operator - Eigenvalues and eigenvectors of an operator: eigenvalues and invariants of powers of an operator, eigenvalues and eigenvectors for symmetric operators, diagonalization of an operator, Cayley-Hamilton theorem, relations between invariants and derivatives of principal invariants in the case n = 3 - Tensor product: semi-Cartesian representation of an operator, eigenvalues and eigenvectors of a tensor product in the case n = 3 - Operators defined in sign: Sylvester criterion, square root operator of a positive definite operator - Polar theorem3. Deformation, kinematics and forces acting on a continuous body (0,7 UFCs)Configuration of a continuum - Deformation gradient operator - Operators of deformation – Operator of inverse deformation - Coefficients of linear, surface and volume expansion – Incompressible bodies – Homogeneous deformation - Small deformations - Velocity and acceleration - Velocity gradient operator - Forces in a continuum - Stress tensor and Cauchy's theorem4. Balance laws and general constitutive principles of continuum mechanics (1.3 UFCs)Law of conservation of mass: Lagrangian and Eulerian formulations - Cardinal equations: boundary conditions - Principle of virtual work - General balance laws: transport theorem, balance law of energy, balance laws of thermo-mechanics in Eulerian form, Galilean invariance (optional), Lagrangian formulation of balance laws, balance law of momentum in Lagrangian form and the first tensor of Piola-Kirchhoff, boundary conditions in Lagrangian variables, law of energy balance in Lagrangian variables - Physical interpretation of the Piola-Kirchhoff tensor, second Piola-Kirchhoff tensor, power of the internal forces in terms of Piola-Kirchhoff tensors - Examples of the Cauchy stress tensor: pressure, simple tension, simple shear - General principles for constitutive laws: the principle of material indifference, the principle of entropy5. Elasticity and thermoelasticity. Fluids. Rigid conductor of heat (1.5 UFCs)Elastic bodies: the consequences of the principle of indifference in the case of elastic material – Thermo-elastic bodies: the principles of material indifference in thermoelasticity, the field equations of thermoelasticity, consequences of the entropy principle in thermoelasticity, isotropic materials - Principle of dissipation in elasticity: one-dimensional nonlinear elasticity - Linear elasticity: isotropic linear elasticity equations - Ideal fluids and Euler equations: boundary conditions in the case of ideal fluids, the work of the internal forces in an ideal fluid – Dissipative fluids of Fourier-Navier-Stokes - Principle of entropy for a fluid - Some special cases of fluids: ideal gases, incompressible fluids of Fourier-Navier-Stokes, compressible fluids of Euler, incompressible ideal fluids and theorem of the three heights – Fluid equations in the Lagrangian formulation - Heat equation – Maxwell - Cattaneo equation Books Suggested text books1. M. Fabrizio: La Meccanica Razionale e i Suoi Metodi Matematici, Zanichelli, Bologna, 19942. T. Ruggeri: Introduzione alla Termomeccanica dei Continui, Monduzzi editore, Bologna, 2008Other references3. S. Bressan, A. Grioli: Esercizi di Meccanica Razionale, Edizioni Libreria Cortina, Padova, 19854. F. John: Partial Differential Equations, Springer-Verlag, Berlin, 19825. V. Smirnov: Corso di Matematica Superiore, Vol.IV/II, Edizioni MIR Editori Riuniti, Roma, 1985 Traditional teaching method Yes Distance teaching method No Mandatory attendance No Written examination evaluation No Oral examination evaluation No Aptitude test evaluation No Project evaluation No Internship evaluation No Evaluation in itinere No Practice Test No

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