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Classical Mechanics Classical Mechanics Jose Antonio Garate Life Sciences Foundation Applied statistical mechanics 2015 [email protected] November 22, 2017 Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 1 / 55 History 300 BC, Aristotelian physics: general principles of change that govern all natural bodies. "continuation of motion depends on continued action of a force". 6th-14th centuries, The theory of impetus. Intellectual precursor to the concepts of inertia, momentum and acceleration in classical mechanics. 17th century Isaac Newton: The three laws of motion " Philosophiae Naturalis Principia Mathematica". 17th century Isaac Newton and Gottfried Leibniz: Calculus. 18th century, Leonhard Euler: Rigid Body Motion. 18th century, Joseph Louis Lagrange: Lagrangian Mechanics. 19th century, William Rowan Hamilton: Hamiltonian Mechanics. 20th century, Relativity and Quantum Mechanics. Classical physics defines the non-relativistic, non-quantum mechanical limit for massive particles. 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 velocity 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 Energy or capacity to do work. 1 2 p2 K = 2 mv or 2m Kinetic Energy. Jose Antonio Garate (Dlab) Classical Mechanics November 22, 2017 3 / 55 The Newton Laws dv 1st Law: No external forces: 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 Time and space 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 ≡ jFjjLj 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 scalar function V (r1; :::; rN), known as a potential energy function, via F(r1; :::::; rN ) = −∇i V (r1; :::::rN ) (5) where ri = @=@ri . Consider the work done by the force Fi in moving particle 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 matter. 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 friction 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 Harmonic oscillator: k x¨ = − x (12) m Z 1 V (x) = − −kx = kx2 (13) 2 Let's propose x(t) = ei!t @ @ei!t k = i 2!2ei!t = − x(t) (14) @t @t m with ! = 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(!t + φ) (15) or p(0) x(t) = x(0) cos(!t) + sin(!t) (16) !m p(t) = p(0)cos(!t) − m!x(0) sin(!t) (17) Assuming energy conservation: p(t)2 1 + m!2x(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 dU dU P V_ = x_ + y_ + ::: = −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 Generalized coordinates qi : cartesian, polar, orientation etc... t2 What to minimize? The action. The integrand in equation (21) is called the Lagrangian L(_q; q). t1 Z t2 X 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 @V(x1−x2) p_ = − , 1 @x1 @V(x1−x2) p_ = − =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.
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