Advanced mechanics

1. Advanced mechanics Physics 302

2. Instructor: Dr. Alexey Belyanin http://faculty.physics.tamu.edu/belyanin/ Office: MIST 426 Office Phone: (979) 845-7785 Email: belyanin@tamu.edu Office Hours: any time when I am in the office

3. 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

4. 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

5. 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

6. PhilosophiæNaturalis Principia Mathematica1687 Herzogin Anna AmaliaBibliothek Weimar

7. 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.

8. 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: Laws of physics are the same in all inertial frames. 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: 6-parameter translations and rotations in space, translations in time, and velocity boosts v = v’ + V, x = x’ + Vt This invariance holds only in closed systems.

9. 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

14. 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

15. 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?