Geometric Algebra in 2D – Fundamentals and Another Look at Complex Numbers (Mathoma)

Geometric Algebra in 3D – Fundamentals (Mathoma)

Geometric Algebra – 3D Rotations and Rotors (Mathoma)

Introduction to Geometric (Clifford) Algebra. (Peeter Joot)

Geometric Algebra: dot, wedge, cross and vector products. (Peeter Joot)

Quadric geometric algebra Top #23 Facts (Rishika Janaki)

Clifford Algebra (David Foster)

Introduction to Clifford algebra (Jose Vargas, Alterman Summer school 31 July 2017)

Algebras de Clifford (Pierre Bayard, Facultad de Ciencias, UNAM)

Clifford Algebra and the Maxwell System: Integral Representation Formulas… (Clifford Algebra and the Maxwell System: Integral Representation Formulas and Radiation Conditions, Emilio Marmolejo, Ciencias TV)

QFT2017 Fall Lecture2.5 – Clifford Algebra (Y-T. Huang at NTU, 17 Sep. 2017, Chinese)

New as playlists:
Space Group Visualizer (Eckhard Hitzer, ICCA8)

Tutorial on Clifford’s Geometric Algebra, (Eckhard Hitzer, Nagoya 2012)

Geometric Algebra for Freshman Course (Eckhard Hitzer, Tokyo 2015)

also in important solutions are although many properties and precised by prof dr mircea orasanu and prof drd horia orasanu ,in multiple relations followed as followed for non holonomic problem with note that for Newtonian equations this matrix is I . The function H in this case is the Legendre transform of L , i.e.
H(p,q) = [p – L(p, )]
(25)
where the infimum is taken over all p . One can show that L is a Legendre transform of H as well, i.e
L(q, ) = [p – H(p,q)].
(26)
In particular, the minimum is attained for
= .
Liouville’s theorem for non-Hamiltonian systems
The equations of motion of a system can be cast in the generic form
where, for a Hamiltonian system, the vector function would be
and the incompressibility condition would be a condition on :
A non-Hamiltonian system, described by a general vector funciton , will not, in general, satisfy the incompressibility condition. That is:
Non-Hamiltonian dynamical systems are often used to describe open systems, i.e., systems in contact with heat reservoirs or mechanical pistons or particle reservoirs. They are also often used to describe driven systems or systems in contact with external fields.
The fact that the compressibility does not vanish has interesting consequences for the structure of the phase space. The Jacobian, which satisfies
will no longer be 1 for all time. Defining , the general solution for the Jacobian can be written as
Thus, we have a conservation law for a modified volume element, involving a “metric factor” . Introducing the suggestive notation , the conservation law reads. This is a generalized version of Liouville’s theorem. Furthermore, a generalized Liouville equation for non-Hamiltonian systems can be derived which incorporates this metric factor. The derivation is beyond the scope of this course, however, the result is
We have called this equation, the generalized Liouville equation Finally, noting that satisfies the same equation as J, i.e.,
the presence of in the generalized Liouville equation can be eliminated, resulting in
which is the ordinary Liouville equation from before. Thus, we have derived a modified version of Liouville’s theorem and have shown that it leads to a conservation law for f equivalent to the Hamiltonian case. This, then, supports the generality of the Liouville equation for both Hamiltonian and non-Hamiltonian based ensembles, an important fact considering that this equation is the foundation of statistical mechanics.
________________________________________
Next: Equilibrium ensembles Up: No Title Previous: Preservation of phase space The Lagrangian
In Lagrangian mechanics we start by writing down the Lagrangian of the system
L = T – U (1)
where T is the kinetic energy and U is the potential energy. Both are expressed in terms of coordinates (q, ) where q is the position vector and is the velocity vector.
he Lagrangian of the pendulum
An example is the physical pendulum (see Figure 1).
Figure 1: The configuration space of the pendulum
The natural configuration space of the pendulum is the circle. The natural coordinate on the configuration space is the angle . If the mass of the ball is m and the length of the rod is l then we have
T = m(l )2
(2)
U = – mglcos
(3)
Thus, the Lagrangian in coordinates ( , ) is
L = m(l )2 + mglcos .
(4)
2.3 Equations of motion
In Lagrangian mechanics the equations of motion are written in the following universal form:
.
(5)
2.4 Pendulum–Equations of motion
For example, for the pendulum we have:
= ml 2 ,
(6)
= – mglsin .
(7)
Thus, the equations of motion are written as
(ml 2 ) = – mglsin .
(8)
This equation can be written as second order equation
ml 2 = – mglsin
(9)
or in the traditional way
= – sin .
(10)
2.5 The meaning of dot
We should emphasize that has dual meaning. It is both a coordinate and the derivative of the position. This traditional abuse of notation should be resolved in favor of one of these interpretations in every particular situation.
2.6 Lagrangian vs. Newtonian mechanics
In Newtonian mechanics we represent the equations of motion in the form of the second Newton’s law:
m = f (q,t)

also in important solutions are although many properties and precised by prof dr mircea orasanu and prof drd horia orasanu ,in multiple relations followed as followed for non holonomic problem with note that for Newtonian equations this matrix is I . The function H in this case is the Legendre transform of L , i.e.

H(p,q) = [p – L(p, )]

(25)

where the infimum is taken over all p . One can show that L is a Legendre transform of H as well, i.e

L(q, ) = [p – H(p,q)].

(26)

In particular, the minimum is attained for

= .

Liouville’s theorem for non-Hamiltonian systems

The equations of motion of a system can be cast in the generic form

where, for a Hamiltonian system, the vector function would be

and the incompressibility condition would be a condition on :

A non-Hamiltonian system, described by a general vector funciton , will not, in general, satisfy the incompressibility condition. That is:

Non-Hamiltonian dynamical systems are often used to describe open systems, i.e., systems in contact with heat reservoirs or mechanical pistons or particle reservoirs. They are also often used to describe driven systems or systems in contact with external fields.

The fact that the compressibility does not vanish has interesting consequences for the structure of the phase space. The Jacobian, which satisfies

will no longer be 1 for all time. Defining , the general solution for the Jacobian can be written as

Thus, we have a conservation law for a modified volume element, involving a “metric factor” . Introducing the suggestive notation , the conservation law reads. This is a generalized version of Liouville’s theorem. Furthermore, a generalized Liouville equation for non-Hamiltonian systems can be derived which incorporates this metric factor. The derivation is beyond the scope of this course, however, the result is

We have called this equation, the generalized Liouville equation Finally, noting that satisfies the same equation as J, i.e.,

the presence of in the generalized Liouville equation can be eliminated, resulting in

which is the ordinary Liouville equation from before. Thus, we have derived a modified version of Liouville’s theorem and have shown that it leads to a conservation law for f equivalent to the Hamiltonian case. This, then, supports the generality of the Liouville equation for both Hamiltonian and non-Hamiltonian based ensembles, an important fact considering that this equation is the foundation of statistical mechanics.

________________________________________

Next: Equilibrium ensembles Up: No Title Previous: Preservation of phase space The Lagrangian

In Lagrangian mechanics we start by writing down the Lagrangian of the system

L = T – U (1)

where T is the kinetic energy and U is the potential energy. Both are expressed in terms of coordinates (q, ) where q is the position vector and is the velocity vector.

he Lagrangian of the pendulum

An example is the physical pendulum (see Figure 1).

Figure 1: The configuration space of the pendulum

The natural configuration space of the pendulum is the circle. The natural coordinate on the configuration space is the angle . If the mass of the ball is m and the length of the rod is l then we have

T = m(l )2

(2)

U = – mglcos

(3)

Thus, the Lagrangian in coordinates ( , ) is

L = m(l )2 + mglcos .

(4)

2.3 Equations of motion

In Lagrangian mechanics the equations of motion are written in the following universal form:

.

(5)

2.4 Pendulum–Equations of motion

For example, for the pendulum we have:

= ml 2 ,

(6)

= – mglsin .

(7)

Thus, the equations of motion are written as

(ml 2 ) = – mglsin .

(8)

This equation can be written as second order equation

ml 2 = – mglsin

(9)

or in the traditional way

= – sin .

(10)

2.5 The meaning of dot

We should emphasize that has dual meaning. It is both a coordinate and the derivative of the position. This traditional abuse of notation should be resolved in favor of one of these interpretations in every particular situation.

2.6 Lagrangian vs. Newtonian mechanics

In Newtonian mechanics we represent the equations of motion in the form of the second Newton’s law:

m = f (q,t)