Download Newtonian mechanics

Advanced mechanics
Physics 302
Instructor: Dr. Alexey Belyanin
http://faculty.physics.tamu.edu/belyanin/
Office: MIST 426
Office Phone: (979) 845-7785
Email: [email protected]
Office Hours: any time when I am in the office
General advices for problem solving:
• Draw a diagram
• Summarize what is given and what you need to find
• Write and solve equations symbolically
• Check units and dimensions
• Check limiting cases
• Check order of magnitude
Minimum math background
• Vector operations: sum, dot product, cross
product, differentiation and integration
• Polar, cylindrical and spherical coordinates
• Differential and integral calculus
• Ordinary differential equations
Classical mechanics
Studies the motion of physical objects
Concepts and mathematical methods are carried on to all
other parts of physics: quantum mechanics, field theory etc.
Newtonian mechanics
Describes this motion based on several assumptions and
postulates that have to be justified by experiment
Philosophiæ Naturalis
Principia Mathematica 1687
Weimar
Herzogin Anna Amalia Bibliothek
Postulates of Newtonian mechanics
A physical object is approximated by a point mass or a system of point masses
Space is three-dimensional and Euclidean.
Positions of particles are defined by their coordinates in 3D space (degrees of
freedom). There are 3 d.o.f. per particle (not necessarily x,y,z).
In closed systems, equations are invariant w.r.t. translations and rotations in
space. (Momentum and angular momentum conversation follow).
Euclidity of space can be checked by measuring distances.
Time is one-dimensional and absolute.
All observers with initially synchronized clocks will measure the same time,
independently on their position and state of motion.
Choice of t = 0 is arbitrary, i.e. equations are invariant w.r.t. time translations.
Energy conservation follows.
Postulates of Newtonian mechanics continued
Galilean Principle of Relativity
There exists a special class of reference frames, called inertial frames, which
have the following properties:
1) Laws of physics are the same in all inertial frames.
2) All reference frames in uniform linear motion with respect to each other
are inertial.
Principle of relativity connects geometry and dynamics
Equations (such as Newton’s second law) are valid only in inertial frames
Properties of space-time hold only in inertial frames
All laws of physics in inertial frames are invariant w.r.t. Galilean transformations: 6parameter translations and rotations in space, translations in time, and velocity
boosts v = v’ + V, x = x’ + Vt
This invariance holds only in closed systems.
A bit of history
Concept of force as a vector and static balance: well known in ancient world
Science of motion: deeply flawed.
Aristotle: there exists one privileged reference frame for each object: the one in
which the object is at rest. The force is needed even to move with a constant
velocity
Galilean ship
Postulates of Newtonian mechanics continued
Newtonian determinism
Initial positions and velocity of all particles uniquely determine their motion
and positions at any future moment of time.
This implies that the equation of motion is second-order differential equation
for positions
Particle motion
Kinematic quantities: vectors of position, velocity, acceleration
Any object is characterized by its inertia. The measure of inertia is mass.
The motion is characterized by a certain value of momentum p = mv
A force F as a cause of change in motion
Newton’s second law
Newton’s first law is not the derivative from the second law
First law allows you to determine if your ref. frame is inertial
How to measure masses? Origin of masses?