Lagrangian Mechanics is the reformulation of Newtonian Mechanics that utilizes the Lagrangian defined by
L=T-U
where
T= total kinetic energy of a system of particles
U= sum of the potential energy functions of a system of particles
In other branches of physics, the Lagrangian is defined as the function
L: TQ--> R
Where Q is the configuration space, a subset of R^3N
such that the action, defined as the functional,
A(q)= int(L) dt
when it reaches stationary value at {q(t)}, will male {q(t)} the equations of motion.
Note, int() means the integration notation with the limits of integration being positive an negative infinity, respectively.
It is shown, using the Calculus of Variations, that the equations of motion are in such a way that they satisfy the Lagrange's Equations of motion
D(d(L)/dv)-d(L)/dq=0
Where D is the differential operator with respect to time, d/dx stands for partial differentiation, and v is the generalized velocity.
L=T-U
where
T= total kinetic energy of a system of particles
U= sum of the potential energy functions of a system of particles
In other branches of physics, the Lagrangian is defined as the function
L: TQ--> R
Where Q is the configuration space, a subset of R^3N
such that the action, defined as the functional,
A(q)= int(L) dt
when it reaches stationary value at {q(t)}, will male {q(t)} the equations of motion.
Note, int() means the integration notation with the limits of integration being positive an negative infinity, respectively.
It is shown, using the Calculus of Variations, that the equations of motion are in such a way that they satisfy the Lagrange's Equations of motion
D(d(L)/dv)-d(L)/dq=0
Where D is the differential operator with respect to time, d/dx stands for partial differentiation, and v is the generalized velocity.
Bob: Why is general relativity so tough to learn?!
Doug: Cause' you don't know enough Lagrangian Mechanics!
Doug: Cause' you don't know enough Lagrangian Mechanics!
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"She could have choked down my entire length, but her mechanical stop left two inches of bare shaft visible."
by Greg T. James June 8, 2005
Get the Mechanical Stop mug.A branch of physics in which the measurable qualities (position, momentum, energy) of a system are described by "observables." The system, upon measurement by the given observable, after being in a superposition of multiple states, collapses into one of a variety of states allowable by the Schrodinger equation. The exact state the system collapses into, is probablistically determined.
X|s1>=x1|s1>
In this example, |s1> is the system being "measured" by the observable X (X denoting the position observable, allowing us to see where the system is within space.) We obtained |s1> as a single state, from a previously superposed (|s> = |s1>+|s2>+|s3>+...)state through an earlier measurement, which collapsed |s> into |s1>, and allowed us to measure the value x1 for its position. In quantum mechanics, all of this information is encoded in the language of vector spaces, as well as linear algebra, and expanded into whats known as a hilbert space (can be thought of as an infinite dimensional vector space).
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In this example, |s1> is the system being "measured" by the observable X (X denoting the position observable, allowing us to see where the system is within space.) We obtained |s1> as a single state, from a previously superposed (|s> = |s1>+|s2>+|s3>+...)state through an earlier measurement, which collapsed |s> into |s1>, and allowed us to measure the value x1 for its position. In quantum mechanics, all of this information is encoded in the language of vector spaces, as well as linear algebra, and expanded into whats known as a hilbert space (can be thought of as an infinite dimensional vector space).
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