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Classical

Jose Antonio Garate

Life Sciences Foundation Applied 2015 [email protected]

November 22, 2017

Jose Antonio Garate (Dlab) November 22, 2017 1 / 55 History 300 BC, Aristotelian : general principles of change that govern all natural bodies. ”continuation of depends on continued of a ”. 6th-14th centuries, The of impetus. Intellectual precursor to the concepts of , and in classical mechanics. 17th century : The three laws of motion ” Philosophiae Naturalis Principia Mathematica”. 17th century Isaac Newton and Gottfried Leibniz: Calculus. 18th century, : Motion. 18th century, Joseph Louis Lagrange: . 19th century, : . 20th century, Relativity and . Classical physics defines the non-relativistic, non-quantum mechanical limit for massive .

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 2 / 55 Some nomenclature

r(t) = (x(t), y(t), z(t)) position vector dr(t) v(t) = dt vector dr(t) p(t) = m dt momentum vector dv(t) d2r a(t) = dt or dt2 acceleration vector F = Force vector. v(t) = r˙(t), a(t) = ¨r(t) over dot notation V, U = Potential or capacity to do . 1 2 p2 K = 2 mv or 2m .

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 3 / 55 The Newton Laws

dv 1st Law: No external : v = C, dt = 0 Inertia. dv 2nd Law: F = ma, where a = dT . FAB = −FBA Action and Reaction. Some comments: These laws assume that the response to a force is instantaneous (no lag). Measurement does not affect system and uncertainty principle does not apply. No QM and lie outside physical existence and are absolute. Different observers can always measure the same time and space. No relativity.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 4 / 55 Newton’s second Law can be restated:

F ¨r(t) = (1) m Equation (1) is a 2nd order differential equation, thus two initial conditions need to be specified.

ZZ F r(t) = dt (2) m For a constant force:

F r(t) = ∆t2 + C ∆t + C (3) m 1 2

C1 = Initial Velocity and C2 =Initial position. Equation (3) uniquely specifies the motion of an object in time.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 5 / 55 Basic Concepts: Mechanical work

Z B WA→B (path) ≡ F · dL (4) A

F θ dl

P *Dot product F · dL ≡ |F||L| cos θ or Fi · Li . It is the projection of a vector into another.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 6 / 55 Basic Concepts: Conservative Forces

Conservative forces are defined as vector quantities that are derivable from a function V (r1, ..., rN), known as a function, via

F(r1, ....., rN ) = −∇i V (r1, .....rN ) (5)

where ∇i = ∂/∂ri . Consider the work done by the force Fi in moving i from points A to B along a particular path. The work done is:

Z B Z B WA→B = Fi · dL = −∇i V · dL = ∆VB→A (6) A A Thus, we conclude that the work done by conservative forces is independent of the path taken between A and B. It follows, therefore, that along a closed path: I Fi · L = 0 (7)

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 7 / 55 Basic Concepts: Newton’s Laws for many particle systems

Classical mechanics in the microscopic world, deals with a large number of particles that are the constituent of . It is assumed that the laws of classical physics can be applied at the molecular level:

Fi = Fi (r1....., rn, r˙i ) (8)

Fi depends on the positions of the rest of the particles plus a term. Now if the force only depends on the individual terms we say that is pairwise additive:

X ext Fi (r1....., rn, r˙i ) = Fi,j (ri − rj ) + F (ri , r˙j ) (9) j6=i Newton’s second law:

mi¨ri = Fi (r1....., rn, r˙i ) (10)

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 8 / 55 Basic Concepts: Phase Space

-6N dimensional space, each point compromises the 3N positions + the 3N momentum

X = (r1...... r3N , p1...... p3N ) → phase space vector (11)

All information required to propagate the system in time is contained in the phase space vector. Classical motion can be visualized a the motion of a phase-space point in time. For N ∼ 1023 it is a enormous element.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 9 / 55 Basic Concepts: Phase Space Examples -1D free particle:

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 10 / 55 Basic Concepts: Phase Space Examples

-1D :

k x¨ = − x (12) m Z 1 V (x) = − −kx = kx2 (13) 2 Let’s propose x(t) = eiωt

∂ ∂eiωt k = i 2ω2eiωt = − x(t) (14) ∂t ∂t m with ω = pk/m and employing euler’s identity eikx = cos kx + i sin kx and subject to two initial conditions:

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 11 / 55 x(t) = A cos(ωt + φ) (15) or

p(0) x(t) = x(0) cos(ωt) + sin(ωt) (16) ωm

p(t) = p(0)cos(ωt) − mωx(0) sin(ωt) (17) Assuming energy conservation:

p(t)2 1 + mω2x(t)2 = C (18) 2m 2

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 12 / 55 Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 13 / 55 Conserved Quantities from Newton’s equations

Let’s assume N particles in 2D: ET = K(v) + V(x) K = 1 m (v 2 + v 2 ) + ..... and K˙ = P 1 2v v˙ = P mv a + ..... 2 1 x1 y1 2 i1 i1 i1 i1 V˙ = dU x˙ + dU y˙ + ... = P −m v a + ..... dx1 1 dy1 1 i i1 i1 ˙ ˙ ˙ X X Et = V + K = mi vi1 ai1 + ..... + −mi vi1 ai1 + ..... = 0 (19) Total Energy is conserved!!!!

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 14 / 55 Conserved Quantities from Newton’s equations

Let’s assume 3 particles in 3D:

F32 3 2 F d 23 Fi = dt (mi vi ) = p˙i F1 = F21 + F31 F 12 F = F + F 31 F 2 12 32 F 21 F 13 F3 = F13 + F23

1 Given that Newton’s 3rd law implies Fij = Fji X X p˙ t = p˙i = Fi = 0 (20) Total momentum is conserved!!!

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 15 / 55 Lagrangian Mechanics Newton’s equations are a local approach, i.e. derivatives, in other words local equations along a trajectory. A global approach only looks at the end-points. There is a unique trajectory that connects both end-points. There is a quantity that is minimized along the trajectory.

t2

What to minimize? Trajectory is a multivariate function. t1

X

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 16 / 55 Lagrangian Mechanics: The Action

Z t2 A = dtK(q ˙ ) − V(q1....qn) (21) t1

t2

Generalized coordinates qi : cartesian, polar, orientation etc... What to minimize? The action. The integrand in equation (21) is called t1 the Lagrangian L(q ˙ , q). X

Z t2 A = dtL(q ˙ , q) (22) t1

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 17 / 55 Lagrangian Mechanics: Calculus of variations

Small first order variation on the t2

trajectory Perturbed trajectory q' 0 q1(t)...... qn(t) → qi (t) + αfi (t) = qi (t) α is any number.

and fi (t) vanishes at the end-points. We postulate that the action is minimized in q0(t) t1 ∂A(α) = 0 (23) ∂α

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 18 / 55 ∂q0(t) ∂q˙ 0(t) i i ˙ ∂α = fi (t), ∂α = fi (t) n ∂A Z t2 X  ∂L ∂L  = dt · fi + · f˙i = 0 (24) ∂α ∂qi ∂q˙ i t1 i=1

Integrating by parts the second term and using the fact that fi vanishes at the end-points:

n ∂A Z t2 X  ∂L d ∂L  = dt fi − = 0 (25) ∂α ∂qi dt ∂q˙ i t1 i=1

Given that dt is not zero and fi only vanishes at the end-points, it implies: ∂L d ∂L − = 0 (26) ∂qi dt ∂q˙ i Equation (26) is known as the Euler-Lagrange equation. There are n equation for each qi generalized coordinate.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 19 / 55 Euler-Lagrange equation

∂L Πi = = The conjugate momentum to qi . ∂q˙ i ∂L = The generalized force. ∂qi Example: Single particle 1D 2 pi L = 2m − V(xi ) ∂L = pi ∂q˙ i ∂L = − ∂V ∂xi ∂xi ∂V p˙ i = − = fi = mi ai ∂xi Thus we recover Newton’s 2nd law.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 20 / 55 Lagrangian and Conservation Laws -Multiple Particle system in Cartesian coordinates:

n 2 X pi ∗ L = − V(x1.....xn) (27) 2mi i=1 * Inter-particle potential. Example: Two particle system in 1D. 2 P2 pi L = i=1 2m − V(x1 − x2) d ∂L = ∂L dt ∂x˙i ∂xi x1 x2 p˙ = − ∂V(x1−x2) , 1 ∂x1 p˙ = − ∂V(x1−x2) =p ˙ +p ˙ = 0 2 ∂x2 1 2 Total momentum is conserved. If we translate the system the Lagrangian does not change (V only depends on distance. Momentum is conserved due to translational symmetry. Relation between symmetries and conservation laws.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 21 / 55 -1 Particle in 2D under gravitation:

V = mgy y 2 p2 L = px + y − mgy 2m 2m Gravitation ∂L p˙ x = ∂x = 0 ∂L p˙ y = ∂y = −mg

x Gravitation breaks translational symmetry in the y dimension. Any change in the y direction changes the Lagrangian.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 22 / 55 -Symmetries and conservations laws are related to operations that do not alter the Action. P ∂F - F (α), dF = dαi = 0 ∂αi -Small variations in the trajectory, first order. qi → qi +  · fi (q), dqi = fi (q) The variation in the action is:

n Z t2 X  ∂L ∂L  dA = dt dqi + dq˙ i = 0 (28) ∂qi ∂q˙ i t1 i=1 Integrating by parts equation (28):

n Z t2   X ∂L d ∂L ∂L t2 dA = dt dqi − + dqi |t1 (29) ∂qi dt ∂qi ∂q˙ t1 i=1

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 23 / 55 Pn ∂L t2 Now, the end-point contributions i=1 ∂q˙ dqi |t1 do not vanish and by virtue of the Euler-Lagrange equation and the condition of null variation of the action dA = 0 the following expression is obtained:

n n X ∂L X dq |t2 = Π f |t2 = 0 (30) ∂q˙ i t1 i i t1 i=1 i=1 Pn In other words the quantity i=1 Πi fi is conserved when performing a Pn variation fi that does not alter the Action. i=1 Πi fi is know as Noether charge.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 24 / 55 Example: Translate all particles by  in the x direction:

t t2 t2 dxi = , fx = 1 qi dxi = 0, fy = 0 qi+ϵfi dz = 0, f = 0 i z t1 t1 q X X Πi fi = pix = C (31) Momentum is conserved due to translational invariance.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 25 / 55 Example: Rotate all particles by  in the x-y plane:

y

dx = −y, fx = −y

dy = x, fy = x ϵ

x X X Πi fi = −px y + py x = L = (32) Angular momentum is conserved due to rotational invariance.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 26 / 55 Example: Move forward in time by  :

T A

q(t) → q(t − ) tA dq dq(t) = − dt  = −q˙  B tB q What is the Variation in the Action:

n Z tA X  ∂L ∂L  dA = dt dqi + dq˙ i + A − B = 0 (33) ∂qi ∂q˙ i tB i=1

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 27 / 55 Integrating by parts equation (33):

n Z tA X  ∂L d ∂L  ∂L dA = dt dq − + dq |tA + A − B = 0 (34) i i tB ∂qi dt ∂qi ∂q˙ i tB i=1 Applying the Euler-Lagrange equation:

n X ∂L t2 dqi |t1 + A − B = 0 (35) ∂q˙ i i=1

And given than  is small, A and B can be approximated by L(tA) ∗  and L(tB ) ∗  respectively.

n X ∂L dq |tA + [L(t ) − L(t )] = 0 (36) i tB A B ∂q˙ i i=1 " n # X ∂L  − q˙ + L |tA = 0 (37) i tB ∂q˙ i i=1

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 28 / 55 Pn ∂L Thus the quantity − q˙ i + L is conserved due to i=1 ∂q˙ i time-translational symmetry and it is defined as −H. H is known as the hamiltonian. X H = q˙ Πi − L(q, q˙ ) (38) Example: 1D Particle in a Potential. 1 2 L = 2 mx˙ − V(x) Π = mx˙ Πx ˙ = mx˙ 2 2 1 2 H = mx˙ − [ 2 mx˙ − V(x)] 1 H = mx˙ 2 + V(x) (39) 2 Thus the total Energy is the quantity that is conserved due to time-translational symmetry. This is the most general definition of energy.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 29 / 55 Hamiltonian Mechanics Laws of mechanics: Subject to conservation Laws. Information is conserved. Nothing disappear or appears. For each degree of freedom we need to know ”where we are” and ”where we are going”

y

x

z

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 30 / 55 Lagrangian : 2nd -order differential equation. 2 d xi m dt2 = Fi -Mathematically it is trivial to convert a 2nd order differential equation in to two first-order differential equations:

dxi m dt = pi dpi m dt = Fi Thus we need position and momentum for each degree of freedom.

p

q

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 31 / 55 The Legendre transform

Consider a function E(V ) of one variable. The Legendre transform allows ∂E to replace the independent variable V by the derivative ∂V . P ∂E(V ) ∂V = P(V ) ∂H(P) P' ∂P = V (P) 0 E(V ) = R V P(V )dV 0 H(P) R P0 E(V) H(P) = 0 V (P)dP H(P) + E(V ) = P0V 0 V' V

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 32 / 55 The Legendre transform II

We define the Legendre transform of the function E(V ) as

H(P) = PV − E(V ) (40)

Thus for any function f (x(y)) the Legendre transform f ∗(y(x)) is defined as: The Legendre Transform ∂f (x(y)) f ∗(y(x)) = x(y) − f (x(y)) (41) ∂x or ∂f (x(y)) −f ∗(y(x)) = f (x(y)) − x(y) (42) ∂x

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 33 / 55 Hamiltonian Mechanics and Legendre transforms

P Based on the definition of the Hamiltonian H = q˙ Πi − L(q, q˙ ) it is evident that it has the form of a legendre transform of the Lagrangian. Let’s compute the variation in H:

X X X ∂L X ∂L dH = q˙ i dpi + pi dq˙ i − dqi − dq˙ i (43) dqi dq˙ i X X ∂L dH = q˙ i dpi − dqi (44) dqi

Thus H is function of the pi ’s and qi ’s. H(q, p)

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 34 / 55 Derivating with respect to each qi and pi equation (44) while keeping the rest of variables fixed we obtain:

∂H =q ˙ i (45) ∂pi qi and

∂H − =p ˙ i (46) ∂qi pi Equations (45) and (46) are know as the Hamilton equation of motion.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 35 / 55 p

 ∂H  =q ˙ i . Velocity of ∂pi qi positions.  −∂H  =p ˙ i . Velocity of ∂qi pi momentums

q

A pair of equations for each degree of freedom that define a flux in phase-space.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 36 / 55 -Energy Conservation

dH X ∂H ∂H  = p˙ + q˙ (47) dt ∂p ∂q

dH X  ∂H ∂H ∂H ∂H = − + = 0 (48) dt ∂p ∂q ∂q ∂p The hamiltonian is conserved!!!

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 37 / 55 General form of conservation Laws

Any property of a system of particles is a function of the phase-space variable p and q. dA(p,q) P h ∂A ∂A i dt = ∂p p˙ + ∂q q˙ dA P h ∂A ∂H ∂A ∂H i dt = ∂q ∂p − ∂p ∂q = {A, H} P h ∂A ∂B ∂A ∂B i Poisson Bracket = {A, B} = ∂q ∂p − ∂p ∂q

dH X  ∂H ∂H ∂H ∂H = − + = 0 (49) dt ∂p ∂q ∂q ∂p -Hamiltonian generates time-depedence P h ∂q ∂H ∂q ∂H i q˙ = {q, H} = ∂q ∂p − ∂p ∂q =q ˙ P h ∂p ∂H ∂p ∂H i p˙ = {p, H} = ∂q ∂p − ∂p ∂q =p ˙

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 38 / 55 Phase-Space incompressibility

Flow in x-y plane Y All fluid moves with the same velocity no gradient of velocity dv Number of points that enter or leave are dy ΔY

the same. dv dx ∂V ∆y + ∂V ∆x = 0 ∂x ∂y ΔX h i ∂V + ∂V = ∇ · F (x ...x ) = 0 ∂x ∂y xi 1 n X

∇xi · F (x1...xn) is defined as the divergence operator. When the fluid velocity (flux) has zero divergence it is called an incompressible fluid.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 39 / 55 X˙ = ρ(X) Phase-space velocity vector

X ∂p˙ i ∂q˙ i X ∂ H ∂ H ∇ · X˙ = + = − + = 0 (50) ∂pi ∂qi ∂pi ∂qi ∂qi ∂pi

The divergence of phase-space flow is incompressible!!!!!!

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 40 / 55 The ensemble distribution function The trajectory approach involves the knowledge with a infinite precision of all p and q in phase-space, in this way is an idealization specially for microscopic systems. We can tackle with employing an statistical approach: Phase Space : All possible microstates available to a system of N particles. Ensemble: Contains all microstates consistent with a set of macroscopic variables e.g Total energy, Volume and number of particles. Thus it is: A Strict subset of all possible phase-space points or Clustered more densely in certain regions of phase-space and less densely in other. The objective is to find the precise mathematical form of how these systems are distributed in phase-space at any point in time.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 41 / 55 Let’s define ρ(X, t) as the ensemble distribution function: The ensemble distribution function: properties

ρ(X, t) ≥ 0 (51)

Z dXρ(X, t) = 1 (52) V It is a density function !!!

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 42 / 55 Bundle of trajectories in a volume element dXt centered around a trajectory Xt All evolving at the same according to Hamilton . How this bundle is distributed in time t?

p p

dXt dX0 dt

q q

We will derive an equation for the time-evolution of ρ(X, t)

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 43 / 55 Phase-space incompressibility. No sources or sinks for any volume element. Members (bundle ) remain constant.

The latter implies that in any volume element Ω in phase-space with a surface S, the rate of decrease (or increase) of ensemble members in Ω must equal the rate at which ensemble members leave (or enter) Ω through the surface S.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 44 / 55 Fraction of phase-space points in volume of phase-space Ω at time t Z dXt ρ(X, t) (53) Ω Rate of decrease of phase-space points:

d Z Z ∂ − dXt ρ(X, t) = − dXt ρ(X, t) (54) dt Ω Ω ∂t

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 45 / 55 The flux of phase-space points through a surface S. This is the number of ensemble members per unit area, per unit time passing through the surface S.

n dS

Z Z ˙ ˙ dSXt · nˆρ(Xt , t) = dXt ∇Xt · Xtρ(Xt , t) (55) S Ω The flux is expressed as a fraction of ensemble members. The right side follows from the divergence theorem.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 46 / 55 Now equating both equations: Z Z ∂ ˙ − dXt ρ(X, t) = dXt ∇Xt · Xtρ(Xt , t) (56) Ω ∂t Ω The choice of the volume element Ω is arbitrary, thus we can equate both integrands and after rearrangement :

∂ ρ(X, t) + ∇ · X˙ ρ(X , t) = 0 (57) ∂t Xt t t

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 47 / 55 now:

˙ ˙ ˙ ∇Xt · Xtρ(Xt , t) = Xt · ∇Xt ρ(Xt , t) + ρ(Xt , t)∇Xt · Xt (58) and from Hamilton equations:

˙ ∇Xt · Xt = 0 (59) Finally we get :

∂ ρ(X , t) + X˙ · ∇ ρ(X , t) = 0 (60) ∂t t t Xt t The last equations defines a total time derivative and it named Liouville’s equation

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 48 / 55 The Liouville equation dρ(X , t) ∂ t = ρ(X , t) + X˙ · ∇ ρ(X , t) = 0 (61) dt ∂t t t Xt t Some comments: ρ(X, t) is a conserved quantity, from LE. Volume in phase-space is conserved from HE Thus:

ρ(X0, 0)dX0 = ρ(Xt , t)dXt (62)

The fraction of member in any volume element dX is conserved, and ensures that we can perform ensembles averages at any point in time.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 49 / 55 Now we can rewrite the LE, remembering that the Hamiltonian generates the time dependence for any function of p’s and q’s and employing the poisson brackets formalism :

˙ Xt · ∇Xt ρ(Xt , t) = {ρ(Xt , t), H(X, t)} (63)

The Liouville equation: poisson bracket formulation dρ(X, t) ∂ = ρ(X , t) + {ρ(X , t), H(X , t)} (64) dt ∂t t t t

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 50 / 55 Equilibrium solution of the Liouville equation

Z A = ha(X)i = dXρ(X, t)a(X) (65)

no external driving forces: H(X, t) → H(X)

Xt (X, t) → Xt (X) At equilibrium, A has not time dependence, thus ha(X)i has no time dependence, if so ρ(X, t) has not explicit time dependence

∂ρ(X, t) = 0 (66) ∂t By the LE we get :

{ρ(X), H(X)} = 0 (67) Thus the solution for ρX is any function of H(X)!!!!

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 51 / 55 ρ(X) ∝ %(X) (68) and to ensure normalization:

%(X) ρ(X) = (69) Z(X) where Z(X) is defined: Z Z(X) = dXρ(X) (70)

Z(X) is the partition function, and it is the essential quantity in equilibrium statistical mechanics. It measures the number of accessible microstates.

1 Z A = ha(X)i = dX%(X)a(X) (71) Z(X)

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 52 / 55 Summary

Classical equations of motion from Newton’s laws.

I Concepts of Mechanical work I Classical equations of motion for many-particle systems. I Phase space concept. I Conserved quantities from Newton’s laws. Lagrangian form of classical mechanics.

I Action, Lagrangian and generalized coordinates. I Symmetries and conserved quantities.

F Translational Invariance: Momentum is conserved. F Rotational Invariance: angular momentum is conserved. F Time invariance: Energy is conserved. Hamiltonian form of classical mechanics.

I Legendre transform of Lagrangian. I Hamiltonian equations of motion. I Flux in phase space. I Liouville’s equation. I Foundation of statistical mechanics.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 53 / 55 References

Gould and Tobochnik (2010) Statistical and with computer applications http://stp.clarku.edu/notes/ Princeton University Press.

Tuckerman (2010) Statistical Mechanics: Theory and Molecular Simulation Oxford University Press.

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 54 / 55 Ω

Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 55 / 55