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From Molecular to Con/nuum Physics I WS 11/12Emiliano Ippoli/| October, 2011
Classical Mechanics
Wednesday, October 12, 2011
Emiliano Ippoliti
Review
2
Physics • Basic thermodynamics • Temperature, ideal gas, kinetic gas theory, laws of thermodynamics • Statistical thermodynamics • Canonical ensemble, Boltzmann statistics, partition functions, internal and free energy, entropy • Basic electrostatics • Classical mechanics • Newtonian, Lagrangian, Hamiltonian mechanics • Quantum mechanics • Wave mechanics • Wave function and Born probability interpretation • Schrödinger equation • Simple systems for which there is an analytical solution • Free particle • Particle in a box, particle on a ring • Rigid rotator • Harmonic oscillator • Basics • Uncertainty relation • Operators and expectation values • Angular momentum • Hydrogen atom • Energy values, atomic orbitals • Electron spin • Quantum mechanics of several particles (Pauli principle) • Many electron atoms • Periodic system: structural principle • Molecules • Two-atomic molecules (H2+,H2, X2) • Many-atomic molecules
Chemistry...
Informatics...
Mathematics...
Wednesday, October 12, 2011
Emiliano Ippoliti
Review
3
Physics • Basic thermodynamics • Temperature, ideal gas, kinetic gas theory, laws of thermodynamics • Statistical thermodynamics • Canonical ensemble, Boltzmann statistics, partition functions, internal and free energy, entropy • Basic electrostatics • Classical mechanics • Newtonian, Lagrangian, Hamiltonian mechanics • Quantum mechanics • Wave mechanics • Wave function and Born probability interpretation • Schrödinger equation • Simple systems for which there is an analytical solution • Free particle • Particle in a box, particle on a ring • Rigid rotator • Harmonic oscillator • Basics • Uncertainty relation • Operators and expectation values • Angular momentum • Hydrogen atom • Energy values, atomic orbitals • Electron spin • Quantum mechanics of several particles (Pauli principle) • Many electron atoms • Periodic system: structural principle • Molecules • Two-atomic molecules (H2+,H2, X2) • Many-atomic molecules
Chemistry...
Informatics...
Mathematics...
Wednesday, October 12, 2011
Emiliano Ippoliti
Classical Mechanics
4
• In Mechanics one examines the laws that govern the motion of bodies of matter. • Under motion one understands a change of place as a function of time.
• The motion happens under the influence of forces, that are assumed to be known.
Wednesday, October 12, 2011
Emiliano Ippoliti
Fundamentalelements
5
Mechanics deals with entities (point-like masses) that can be completely described by:
• their position, i.e. 3 real numbers at each time:
• a mass m, i.e. a real number that gives a measure of their inertia.
r t( ) = x t( ), y t( ), z t( )( )
In this context the time can be seen as a independent process and therefore a free parameter.
Wednesday, October 12, 2011
Emiliano Ippoliti
Some definitions
6
The velocity is given by
r t( ) = drdt
The acceleration is defined by
r t( ) = d 2rdt 2
=drdt
Wednesday, October 12, 2011
Emiliano Ippoliti
Inertial frames
8
To write down the equations of motion for a certain problem, one first has to choose a frame of reference. The goal is then to find a frame of reference in which the lawsof mechanics take their simplest form.
It can be derived that one can always find a frame of reference in which space is homogeneous and isotropic and time is homogeneous. This is then called an ”inertial frame”. In such a frame a free body which is at rest at some instant of time always remains at rest. The same is true if the body moves with a velocity constant in both magnitude and direction. This is the Newton’s First Law.
Wednesday, October 12, 2011
Emiliano Ippoliti 9
Galileo’s Relativity Principle
Experiment shows that there is not one, but an infinity of inertial frames moving, relative to another, uniformly in a straight line. In all these frames the properties of space and time are the same and so are the laws of mechanics.
This constitutes the”Galileo’s Relativity Principle”
Wednesday, October 12, 2011
Emiliano Ippoliti
Forces
10
The development in time of the position is given by the equations of motions:
mr t( ) = F r t( ), r t( ), …, t( )
where F is a given function that describes the forces acting on the system (this can also be considered the definition of force). This is the Newton’s Second Law.
The basic challenge of the “Newtonian Mechanics” is to find the solution to this equation with the boundary conditions:
r t0( ), r t0( )
Wednesday, October 12, 2011
Emiliano Ippoliti
PotentialEnergy
11
Especially simple and important kind of forces are of the form
F r( ) = −
∇U r( )
where stands for the potential energy. Ur( )
From the equation of motion, inserting this kind of force we get in the coordinate notation:
mri t( ) = −
ddriU ri( )
Wednesday, October 12, 2011
Emiliano Ippoliti
Motion constant:Total Energy
12
Taking the scalar product with this becomes ri t( )
mri t( ) ri t( ) = −
dridt
ddriV ri( )
Using the derivative chain rule:
ddt
mri t( )2( ) = 2mriri ddtU ri t( )( ) = dU
drii∑ dri
dt
we can rewrite our equation as the total differential
ddt
12mr 2 t( ) +U r t( )( )⎛
⎝⎜⎞⎠⎟= 0
Kinetic Energy
Wednesday, October 12, 2011
Emiliano Ippoliti
Principle of Least Action
14
Let the mechanical (n particles) system fulfill the boundary conditions
ra t1( ) = ra1
ra t2( ) = ra2Then, the condition on the system is that it moves between these positions in such a way that the integral
S = L
r1,r2 ,…, r1,
r2 ,…,t( )dtt1
t2∫is minimized. S is called the action and L is the Lagrangian associated with the given system.
Wednesday, October 12, 2011
Emiliano Ippoliti
Lagrange’sEquations
15
Using the variational calculus it can be proved that minimizing the action is equivalent to solve the system of 3n second-order differential equations:
ddt
∂L∂ra
⎛⎝⎜
⎞⎠⎟−∂L∂ra
= 0 a = 1,…,n
These are the ”Lagrange’s Equations” and are the equations of motion of the system (they give the relations between acceleration, velocities and coordinates). The general solution contains 6n constants, which can be determined from the initial conditions of the system (e.g. initial values of coordinates and velocities).
Wednesday, October 12, 2011
Emiliano Ippoliti
Uniqueness
16
If one considers two Lagrangian functions differing only by a total derivative with respect to time of some function f(r,t):
′L r, r,t( ) = L r, r,t( ) + d
dtf r,t( )
one finds that the action
′S = ′L r, r,t( )dt
t1
t2∫ = L r, r,t( )dtt1
t2∫ + f r,t( )dtt1
t2∫ = S + f r2 ,t2( ) − f r1,t1( )
differs only by a quantity which gives zero on variation. This means the conditions δS = 0 and δS′ = 0 are equivalent and the form of the equations of motion is unchanged. Therefore, the Lagrangian is only defined up to an additive total timederivative of any function of coordinates and time.
Wednesday, October 12, 2011
Emiliano Ippoliti
Lagrangian of a free particle
17
The Lagrangian associated to a system has to have the same symmetry properties of the system itself:
- In a free particle system the space is homogeneous
L cannot depend explicitly on the vector position
r
- In a free particle system the time is homogeneous
L cannot depend explicitly on the time t
- In a free particle system the space is isotropic
L cannot depend on the direction of the vector velocity
r
dLdt
= 0 ⇒ L = Lr , r( )
dLdr
= 0 ⇒ L = Lr( )
L = Lr 2( ) = L
v 2( ) = L v2( )
Wednesday, October 12, 2011
Emiliano Ippoliti 18
Lagrangian of a free particle
′L ≡ ′L
v 2( ) = L′v 2( ) = L
v 2 + 2v ⋅ ε + ε 2( ) ≈ Lv 2( ) + ∂L
∂(v2 )2v ⋅ ε
If two frames K and K′ move with an infinitesimal velocity to each other, then
ε
′v = v + ε
But since the equations of motion must have the same form in every frame, the two Lagrangians L(v2) and L′(v2) can only differ by a total time derivative of a function of r and t:
The last term can only be a total time derivative if it is a linear function of the velocity. Therefore, is independentof the velocity, which means that the Lagrangian can be written as
L =
12mv2
r
Wednesday, October 12, 2011
Emiliano Ippoliti
Additivity
19
If a mechanical system consists of two separate parts A and B, each part would have a separate Lagrangian LA and LB
respectively. If these parts do not interact, e.g. in the limit where the distance between the parts become so large that the interaction can be neglected, the Lagrangian of the system is given by
L = LA +LB
This additivity states that the equations of motion of part A cannot be dependent on quantities pertaining to part B and vice versa.
Wednesday, October 12, 2011
Emiliano Ippoliti
Lagrangian of a system of particles
20
To derive the Lagrangian for a system of particles which interact with each other we can start by taking the Lagrangian for the non-interacting particles:
and adding a certain function of the coordinates, depending on the nature of the interaction:
L =
12mava
2
a∑
L =
12mava
2
a∑ − U r1,
r2 , …( )The first part of the Lagrangian we can recognize to be the kinetic energy (T), and the second part to be the potential energy (U).
Wednesday, October 12, 2011
Emiliano Ippoliti
Lagrangian of a system of particles
21
In fact, if we derive the equations of motion for the system by replacing that Lagrangian in the Lagrangian equations:
ddt
∂L∂va
⎛⎝⎜
⎞⎠⎟−∂L∂ra
= 0 a = 1,…,n
we obtain
ma
dvadt
= −∂U∂ra
a = 1,…,n
i.e. the Newton’s equations, if we recognize the second term as the forces acting on the system:
Fa = −
∂U∂ra
Wednesday, October 12, 2011
Emiliano Ippoliti
When Lagrangian Mechanics
22
It is often necessary to deal with mechanical systems in which the interactions between different bodies take the form of constraints. These constraints can be very complicated and hard to incorporate, when working directly with the differential equations of motion.
Here it is often eas ier to construct a Lagrange function out of the kinetic and potential energy of the system and derive the correct equations of motion from there.
Wednesday, October 12, 2011
Emiliano Ippoliti
Hamiltonian Mechanics
23
Hamilton, William Rowan (1805–1865)
Wednesday, October 12, 2011
Emiliano Ippoliti
Generalized coordinates
24
The formulation of classical mechanics in terms of the Lagrangian is based on formulating the state of a mechanical system in terms of coordinates ( ) and velocities ( ). But this is not the only possibility! Sometimes it is more advantageous to choose another set of quantities to describe a mechanical system, e.g. generalised coordinates
ra
and momenta
ra
qa =qara ,ra( )
pa =∂L∂qara ,ra( )
Wednesday, October 12, 2011
Emiliano Ippoliti
Legendre transformation
25
Then the total differential of the Lagrangian is given by
dL =
∂L∂qa
dqaa∑ +
∂L∂qa
dqaa∑ = pad
qaa∑ + pad
qaa∑
From the Lagrangian equation:
ddt
∂L∂qa
⎛⎝⎜
⎞⎠⎟−∂L∂qa
= 0 ⇒∂L∂qa
=dpadt
= pa
Using the differential chain rule applied on the last term
dL = pad
qaa∑ + d pa
qa( )a∑ − qad
paa∑
Wednesday, October 12, 2011
Emiliano Ippoliti
Hamiltonian
26
Rearranging this we end up with
d pa
qaa∑ −L
⎛⎝⎜
⎞⎠⎟= − pad
qaa∑ + qad
paa∑
the argument of the total differential on the left side is the total energy of the conservative system (T+U), now only expressed in terms of coordinates and momenta. In this form we call it the ”Hamiltonian” or ”Hamilton’s function” of the system:
Hq, p,t( ) = pa
qaa∑ −L
qa∂L∂qaa
∑ = paqa
a∑ = 2T
Since the L = T - U and we can write (using Euler’s theorem on homogeneous functions):
Wednesday, October 12, 2011
Emiliano Ippoliti
Hamilton’s equations
27
Then the total differential of the Hamiltonian is given by
dH = − pad
qaa∑ + qad
paa∑
Using this we can derive the required equations of motion in terms of coordinates and momenta:
qa =∂H∂pa
pa =∂H∂qa
Hamilton’s equations: set of 6n first-order differential equations for the 6n unknown functions and
pa (t) qa (t).
Wednesday, October 12, 2011
Emiliano Ippoliti
Lagrange vs Hamilton
28
The difference between the Lagrangian and Hamiltonian formalism is only a Legendre transformation. For some problems it is easier to write L down, for others H.
One is given by a system of 3n second-order differential equations, while the other one describes the same physical problem with 6n first-order differential equations.
Wednesday, October 12, 2011
Emiliano Ippoliti
Summarizing
29
Newtonianmechanics
Lagrangianmechanics
Hamiltonianmechanics
SystemForces Lagrangian Hamiltonian
Equations of motion
Type of mathematical
problem
3n second(or more)-order differential equations
3n second-order differential equations
6n first-order differential equations
L = Lra ,ra ,t( ) H = H
qa ,pa ,t( )
Fa =
Fara ,ra ,ra ,...,t( )
Fa = ma
ra
ddt
∂L∂ra
⎛⎝⎜
⎞⎠⎟−∂L∂ra
= 0
(a = 1,…,n)
(a = 1,…,n)
qa =∂H∂pa
pa =∂H∂qa
Wednesday, October 12, 2011
Emiliano Ippoliti
Venn diagram
30
Under some assumptions, we can represent the sets of the systems that can be described by Newtonian (N), Lagrangian (L) and Hamiltonian (H) mechanics this way:
N L H
Wednesday, October 12, 2011
Emiliano Ippoliti
References
31
1. M. Ganz. Introduction to Lagrangian and Hamiltonian Mechanics. http://image.diku.dk/ganz/Lectures/Lagrange.pdf
2. H. Goldstein. Classical mechanics. Addison-Wesley, Cambridge, 1950.
3. L.D. Landau and E.M. Lifschitz. Mechanics. Butterworth, Oxford, 1981.
Wednesday, October 12, 2011
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