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Physical Thought: Lagrangian Mechanics: From the Principle of Least Action to the Euler-Lagrange Equation

http://physical-thought.blogspot.com/2008/09/lagrangian-mechanics-from-principle-of.html

A blog of physics, mathematics, programming and photography. Sunday, 14 September 2008. Lagrangian Mechanics: From the Principle of Least Action to the Euler-Lagrange Equation. A quantity known as the Action, A is defined as. A = int L ,dt. Where L is known as the Lagrangian. The simplest Lagrangian is given as the kinetic energy T of a system minus the potential energy V of the system:. L(x,t) = T(x,t) - V(x,t). X(t) rightarrow x(t) a(t). V(t) rightarrow v(t) dot{a}(t). A(t 0) = a(t 1) = 0. The Action t...

physical-thought.blogspot.com

Physical Thought: Four-Vector Relativistic Velocity Transformations

http://physical-thought.blogspot.com/2008/04/four-vector-relativistic-velocity.html

A blog of physics, mathematics, programming and photography. Tuesday, 15 April 2008. Four-Vector Relativistic Velocity Transformations. Following on from my earlier post. I decided to write about the relativistic velocity transformations using a four-vector approach. We can define a space-time four-vector (or position-time) as follows:. X mu = left( begin{array}{c} ct x y z end{array} right). The Lorentz transform can be written in matrix form, using. Gamma= frac{1}{ sqrt{1- beta 2}. 0 and 0 and 1 and 0.

physical-thought.blogspot.com

Physical Thought: Gauge Invariance in Quantum Mechanics

http://physical-thought.blogspot.com/2010/03/gauge-invariance-in-quantum-mechanics.html

A blog of physics, mathematics, programming and photography. Tuesday, 2 March 2010. Gauge Invariance in Quantum Mechanics. This post continues the 'Road to Quantum Field Theory' series. The Lorentz force law for a non-relativistic particle of charge q, moving with velocity v in both electric ( E. And magnetic ( B. Fields, is given by:. Mathbf{F} = q mathbf{E} q mathbf{v} times mathbf{B}. This can be derived from Hamilton's equations using the classical Hamiltonian given below. Which replace the operators.

physical-thought.blogspot.com

Physical Thought: Lagrangian Mechanics: From the Euler-Lagrange Equation to Simple Harmonic Motion

http://physical-thought.blogspot.com/2008/09/lagrangian-mechanics-from-euler_20.html

A blog of physics, mathematics, programming and photography. Saturday, 20 September 2008. Lagrangian Mechanics: From the Euler-Lagrange Equation to Simple Harmonic Motion. I already wrote about obtaining Newton's Laws from the Principle of Least Action. Now I'm going to analyse a simple mass-spring system; effectively just a case of substituting in a suitable potential for the spring. T= frac{m dot{x} 2}{2}. Where k is the spring constant (N m -1). So, for our mass-spring system, the Lagrangian is. It is...

physical-thought.blogspot.com

Physical Thought: Symmetries, Conserved Quantities and Noether's Theorem

http://physical-thought.blogspot.com/2008/09/symmetries-conserved-quantities-and.html

A blog of physics, mathematics, programming and photography. Wednesday, 24 September 2008. Symmetries, Conserved Quantities and Noether's Theorem. Following on from my posts on Lagrangian and Hamiltonian Mechanics [ 1. I'll start by showing that the formulation of Lagrangian and Hamiltonian Mechanics, thus far, allows us to determine several conservation laws. Consider, for example, the homogeneity. If we make a transformation. Mathbf{r} rightarrow mathbf{r} delta mathbf{r}. L rightarrow L delta L. Delta...

physical-thought.blogspot.com

Physical Thought: Reversing Lines In A File

http://physical-thought.blogspot.com/2009/07/reversing-lines-in-file.html

A blog of physics, mathematics, programming and photography. Wednesday, 15 July 2009. Reversing Lines In A File. As is often the case, an interesting little text manipulation task came up in the office today; given a file containing lines of text, reverse it (i.e. put the last line first, the penultimate line second, etc.). For line in open(sys.argv[1]):. For line in lines:. After this, I tried to shorten it a bit. Putting the lines into a list could be done nicely with a list comprehension,. See: http:/...

physical-thought.blogspot.com

Physical Thought: Farscape Intro Randomiser

http://physical-thought.blogspot.com/2010/02/farscape-intro-randomiser.html

A blog of physics, mathematics, programming and photography. Monday, 8 February 2010. While watching Farscape Season 2, Episode 4 (Crackers Don't Matter) last night, I wrote a Python script to randomise elements of the Farscape intro. Here's an example:. Posted by Andrew J. Bennieston. Subscribe to: Post Comments (Atom). Andrew J. Bennieston. View my complete profile. Clang (LLVM) C Compiler. Writing Scientific Documents Using LaTeX.

physical-thought.blogspot.com

Physical Thought: An Introduction to Relativity

http://physical-thought.blogspot.com/2008/04/introduction-to-relativity.html

A blog of physics, mathematics, programming and photography. Monday, 14 April 2008. An Introduction to Relativity. Einstein's Principles of Relativity. All inertial frames are equivalent with respect to the laws of physics (Principle of Relativity). The speed of light in empty space is independent of the state of motion of its source (Constancy of the speed of light). Assuming that there is no change of direction of motion, and we have a linear transformation, i.e. X=x prime=0 , t=t prime=0. Delta x) 2 -...

physical-thought.blogspot.com

Physical Thought: Basic Relativity

http://physical-thought.blogspot.com/2008/11/basic-relativity.html

A blog of physics, mathematics, programming and photography. Monday, 10 November 2008. Consider a coordinate frame S' moving with velocity v in the positive x-direction of a frame S. The Lorentz transforms between the two frames (x, y, z, t in S and x', y', z', t' in S') are given by:. X prime &= gamma(x-vt) = frac{1}{ sqrt{1- frac{v 2}{c 2} }(x - vt). Y prime &= y. Z prime &= z. T prime &= gamma left(t - frac{vx}{c 2} right) = frac{1}{ sqrt{1- frac{v 2}{c 2} } left(t - frac{vx}{c 2} right). Y &= y prime.