2.1. The Lagrangian and its symmetries
In classical dynamics, a system’s equations of motion can be derived from its Lagrangian, L; the difference between the kinetic and potential energies of the system, here denoted by T and V respectively. This is often given as a function of a set of generalized coordinates, qi
, and their time derivatives, ˙qi
. This treatment is sufficient to describe a discrete system of particles, however in order to properly describe a continuous system, the Lagrangian must be replaced by a Lagrangian density: L(qi
, q˙i) → L(φi
, ∂µφi). (1)
Whereby the generalized coordinates qi are replaced by the fields φi(x µ ), and the time derivatives are replaced by the derivative of these fields with respect to each of the four space-time coordinates,
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(2)
(It should be noted that henceforth a system’s Lagrangian Density will be referred to only as its
Lagrangian and no reference will be made to a system’s Lagrangian.)
To formulate a Quantum Field Theory, one must replace the single particle wavefunctions of quantum mechanics with multi-particle excitations of a quantum field that must satisfy the field equation derived from the system’s Lagrangian. The Lagrangian for a spinor field ψ(x) satisfying the free-particle Dirac equation is given by [9]
LD = iψγµ
∂µψ − mψψ (3) where γ µ represents the four Dirac gamma matrices, and ψ(x) the four component complex spinor; ψ(x) =
ψ1 ψ2 ψ3 ψ4
, γ0 =
1 0
0
In the book, Theories of Relativity, sixteen year old Dylan calls Brendan a vulture because he bullies weak and defenceless people into doing panhandling for him, steals money off his recruits, and he only recruits homeless confused minors and drugs them to do what he wants. Dylan is smart enough to not follow Brendan as a recruit and the more time he spends on the street watching Brendan he realizes how good of an idea that was because he didn’t become “vultures” pray. One reason Dylan calls Brendan a vulture is because he has seen Brendan bully youths like Jenna into saving her money for himself. One time we see this in the book is when Jenna is in Mandy’s Donut shop and she buys Dylan a meal with her money. After some time with them talking Brendan walks in and sees them both
Kinetic energy is the energy an object possesses due to its motion. The difference between kinetic energy and potential energy is the kinetic energy is the energy of an object that is already in
This will lead to an explanation of motion, the development of the calculus, and the establishment of basic laws of modern physics.
We clash with yellow. Find me tonight under the bleachers.” Newton was wrong in his first and second laws of motion because with the new information and exceeding technology that we have today we can study closer and at a higher intensity than we could during Newton’s time. The first law of motion ,often referred to as the “law of inertia”, violates the Uncertainty Principle which is the principle that the momentum and position of a particle cannot both be precisely determined at the same time. The second law of motion violates the of equation of quantized energy. Quantize means to restrict (a variable quantity) to discrete values rather than to a continuous set of
least five aspects, each one associated with one of the cardinal directions and the center. They
rise to matter and life ‘being driven into closer union and held there by the entanglement of their
Dylan is a tall and lean young man, not even out of high school, life on the streets is hard. He was kicked out by his mother when she found a new man,16-year-old Dylan Wallace has been panhandling on the streets, preparing for winter. Dylan panhandles barely enough money to eat. As the weather gets colder and the going gets harder Dylan wants to know what he did to deserve this life. He doesn’t want to deal drugs or turn tricks like his friends, he does not want to be one of ‘’vulture’’ slaves. Dylan can’t get a job because he has no fixed address, he’s always dirty and hungry his long dark hair often matted his head. He has turned to petty theft. He doesn't trust anyone who wants to help, like Ainsley, a former street kid now working her way through school to become a social worker.
and this quantity of motion is “the product of its speed and its size” (Descartes, 1644/2012, p.33), but the conception of speed used by Descartes is an scalar rather than considered as a vector like velocity (as modern development shows (Gerald Holton and Stephen G. Brush , 2001, p. 210)), nonetheless Descartes suggests that “the mind does not directly move the external limbs, but simply controls the animal spirits which flow from the heart via the brain into the muscles, and sets up certain motions in them..” (Descartes, 1641/1985, p. 161), and this allow the interaction of substances in Cartesian terms, in other words, according to Descartes, the soul only change the direction of the body, but cannot add motion to the physical bodies; however, this is the problem of the interaction in Cartesian terms: as Dennett explain “any change in the trajectory of any physical entity is an acceleration requiring the expenditure of energy” (Dennett, 1991, p. 35), thus, the Cartesian explanation should violate the laws of physics, in other words as Dennett asks “…where is this energy to come from?” (Dennett, 1990, p.35), and this energy cannot come from a
On September 9, 1993 a seventeen year old boy, Christopher Simmons, and his a few of his friends met up to discuss and devise a plan to commit a robbery and maybe even a murder, just for fun. Simmons’ plan was not complicated: find someone to burglarize, tie up the victim and either leave the victim tied to a tree or push them off a bridge. Simmons and his accomplice went through a window that was slightly cracked and proceeded into the bedroom of Mrs. Crook. The two teenagers tried the woman up and loaded her into the back of her minivan. They drove to the state park at the edge of town where they had planned to dispose of the body and that is exactly what they did. The boys were caught later on that month put on trial for murder in
Aristotle’s understanding of the four causes begins with the assumption that is present in all Greek philosophy, the notion of pre-existing matter. He observed the world around him and noticed that it was in a state of constant motion, a movement from potentiality to
One of the strongest points that Laqueur makes in this text is about the true nature
There are many catholic social justices but there are only two social justices that really affected me. The first of these is the social justice of rights and responsibilities. This social justice deals with doing our duty to help and serve our fellow man. The next social justice is care for God's creations. After being assigned this justice to base the organic ion keep the ocean clean I learned the importance of keeping the earth and the creation of it safe, clean and healthy.
Theory is a journey to uncover the past and improve the future. By uncovering and analyzing a discipline’s theoretical journey, insight and self-awareness are gained. According to Meleis (2012), “Theories are reservoirs in which related knowledge is articulated and organized into meaningful wholes” (p.33). By implementing and analyzing theories, empowerment and guidance for the future is obtained.
In Metaphysics XII, Aristotle elaborates on a need for a “first mover that initiates motion without being moved” (Met. 12.7, 1072a26). This primary, or unmoved mover, he believes is the source of all motion in the universe. In this essay, I will explain his conception of such a mover. I will then elaborate on how this unmoved mover initiates motion. Finally, I will explain his rationale for believing there is such a mover.
Prior to the boom of the craft beer industry, if one wanted to imbibe in a nitrogen-infused beer, the choices were limited. The most popular drink of choice was that “black stuff” originating from St. James’ Gate Brewery in Dublin.