Advanced mechanics. Physics 302

Transcription

1 Advanced mechanics Physics 302

2 Instructor: Dr. Alexey Belyanin Office: MIST 426 Office Phone: (979) Office Hours: any time when I am in the office

3 Do your homework! Weekly homework will be assigned and graded Two mid-term tests and the final HW: 20%, two mid-terms: 50%, final: 30% Homework: you can work together, but submit separately 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 We will learn and use: Calculus of variations More differential equations Matrix operations, eigenvalue problems

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. NOT a dead field like Latin language: new developments in chaos, many-body dynamics etc.

6 Newtonian mechanics Describes this motion based on several assumptions and postulates that have to be justified by experiment The most intuitive formulation, but not invariant w.r.t. transformation of coordinates in configuration space or phase space Cannot be generalized to quantum mechanics and relativity Lagrangian and Hamiltonian mechanics: identical in content to Newtonian mechanics, but utilize symmetry of the equations and solve more problems.

7 Herzogin Anna Amalia Bibliothek Philosophiæ Naturalis Principia Mathematica 1687 Weimar Buchenwald

8 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. Triangulation. Checking if the sum of angles is equal to 180.

9 Postulates of Newtonian mechanics 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.

10 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

11 Particle motion Any object is characterized by its inertia. The measure of inertia is mass. A large mass is harder to accelerate. How to measure masses? Be careful with measuring inertial mass, not gravitational mass! Inertial balances. Origin of masses? What if a particle has a small mass but a large velocity? The motion is characterized by momentum p = mv A force F as a cause of change in motion. Its definition includes the way to measure the force.

12 Newton s first and second law If the net force F is zero, a particle moves with a constant momentum p = const. The rate of change of momentum is equal to the net force: dp/dt = F Newton s first law is not the derivative from the second law! First law postulates a special class of reference frames, inertial frames, and allows you to determine if your ref. frame is inertial. You need to measure forces independently. If net force = 0, measure p. If p = const, you are in an inertial frame. Note that the first law does not specify the value of p, only the fact that p = const. Thus all reference frames with p = const are equivalent, and equations of motion should look the same in any of them. The second law is valid only in the inertial frame!

13 The first law was actually formulated by Galileo as a principle of relativity 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. Reference frames are only approximately 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, r = r + Vt This invariance holds only in closed systems.

14 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

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