PHYSICS: A BRIEF SUMMARY MIGUEL A. LERMA 1. Introduction This is a brief introduction to Physics intended for “the impatient”. Its purpose is to give a brief summary of a number of core theories in Physics. Usually it takes several years for a Physics student to learn these theories, but for some practical purposes all you need to know can be told in the time it takes to read a booklet like this one. This work is conceived as a dynamic document, that will be posted on the web and modified periodically to expand some sections, correct possible mistakes, and include further subjects of interest. Look for it at ~mlerma/courses/e11-99s/physics.pdf Please, send me your suggestions—email address at the end. 2. Mechanics 2.1. Newton’s Laws. Ordinary Mechanics is ruled by Newton’s laws. The motion of a particle is described by (2.1) F = m a , where F is the applied force, m is the mass of the particle, and a = d v /dt = d 2 r /dt 2 is the particle’s acceleration, with v being its velocity and r is position vector. In coordinates equation (2.1) looks like this: (2.2) F i = m d 2 x i dt 2 ( i = 1 , 2 , 3) . 2.2. Euler-Lagrange equations. Newton’s law as described above is easy to use in Cartesian coordinates for mechanical problems without constrains, but it can be generalized in a way that makes it easier to apply to more general situations. In one dimension Newton’s law is (2.3) m ¨ x - F ( x, t ) = 0 , Date : July 10, 2013. 1
2 MIGUEL A. LERMA where the dot denotes time derivative. If the force derives from a potential V ( x, t ), then F ( x, t ) = - ∂V ( x, t ) /∂x . On the other hand, by using the kinetic energy T ( ˙ x ) = m ˙ x 2 / 2, and the momentum p = m ˙ x = ∂T/∂ ˙ x we see that m ¨ x = dp/dt , hence (2.4) d dt ∂T ∂ ˙ x + ∂V ∂x = 0 . Now we introduce the Lagrangian function, L ( x, ˙ x ) = T ( ˙ x ) - V ( x ), and the equation becomes: (2.5) d dt ∂L ( x, ˙ x ) ∂ ˙ x - ∂L ( x, ˙ x ) ∂x = 0 . Its generalization to any number of (non necessarily Cartesian) coor- dinates q 1 , q 2 , . . . , q n is the Euler-Lagrange equation: (2.6) d dt ∂L ( q k , ˙ q k , t ) ∂ ˙ q k - ∂L ( q k , ˙ q k , t ) ∂q k = 0 ( k = 1 , 2 , . . . , n ) . It turns out that not only mechanical systems but also many other physical systems can be described by an equation like (2.6) with a suitable Lagrangian L . The choice of Lagrangian is dictated by physi- cal experience, although some authors (such as Landau) have tried to derive it from general principles. 2.3. Hamilton’s Principle. The action of a physical system with a given Lagrangian L ( q k , ˙ q k , t ) between times t 1 and t 2 is defined by the integral (2.7) S ( q k ( t )) = Z t 1 t 0 L ( q k ( t ) , ˙ q k ( t ) , t ) dt. That integral depends on the path q ( t ) followed by the system between t 0 and t 1 . Equation (2.6) turns out to be equivalent to the fact that the action (2.7) is a critical point (usually a minimum) in the space of paths with fixed endpoints q k ( t 0 ) and q k ( t 1 ): (2.8) δS = δ Z t 1 t 0 L ( q k , ˙ q k , t ) dt = 0 .
- Spring '15
- Physics, MIGUEL A. LERMA