# FYSA2020 Mechanics (5 cr)

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## Description

Newtonian mechanics (Newton’s laws and conservation laws)

Oscillations (linear, damped, two-dimensional)

calculus of variations (Euler-Lagrange equations)

Lagrangian mechanics (Lagrangian, generalized coordinates, constraints, connection between conservation laws and symmetries, basics of Hamiltonian mechanics)

gravitation and central-field motion (Newton’s law of gravitation, central-field motion, effective potential, planetary motion)

Newtonian mechanics and selected types of problems

non-intertial coordinate systems (rotating coordinate systems, centrifugal force and Coriolis force)

rigid-body dynamics (energy and angular momentum, moments of inertia, principal axes, equations of motion for rotations, stability of rotations)

coupled oscillations (general procedure of solving, applications)

methods for solving various types of problems (lecture examples and problem sets)

## Learning outcomes

After completing the study unit the student can

describe the foundations of Newtonian mechanics

apply Newtonian mechanics in three-dimensional situations

decide when Lagrangian or Hamiltonian mechanics is particularly useful

apply Lagrangian mechanics in cases with constraints

reduce for central forces a two-body problem into one-body problem

describe planetary motion and the concept of constant of motion

deduce the apparent forces in a rotating corrdinate system

for the moment of inertia tensor for a rigid body

apply the equations of motion of a rigid body

find normal modes and their frequencies for coupled oscillations

solve applied problems staring from the equations of the basic theory

## Description of prerequisites

**Basic physics studies:**

FYSP1010 Introduction to Mechanics

FYSP1020 Waves and thermodynamics

**Selected topics from courses in mathematics:**

MATP211-213 Calculus 1-3, especially:

Properties of elementary functions

Derivative and integral

MATA114 Differential equation; especially:

1st order separable differential equation

2st order constant-coefficient differential equation

MATA181-181 Vector Calculus 1-2, especially:

Differential, partial derivative, gradient

Integral over a three-dimensional volume

Integral over a path

MATP121-122 Linear Algebraand Geometry 1-2; especially

Vectors: sum, dot product, cross product

Matrix and determinant

Products of vectors and matrices

Rotation matrices

Eigenvalues and eigenvectors

In addition the basic properties of complex numbers.

## Study materials

Luentomateriaali TIM:ssä

## Literature

- Thornton & Marion: Classical dynamics of particles and systems. ISBN: 978-0-495-55610-7

## Completion methods

### Method 1

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### Method 2

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### Teaching (5 cr)

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Luennot, laskuharjoitukset, tentti ja laboratoriotyöt.

#### Teaching

##### 3/14–5/22/2022 Lectures

##### 5/20–5/20/2022 Final exam

##### 6/3–6/3/2022 Final exam

### Independent study (5 cr)

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Self-study, exam and laboratory work.