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Lecture I, Aug25, 2014 Newton, Lagrange and Hamilton's Equations of Classical Mechanics

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Lecture I, Aug25, 2014 Newton, Lagrange and Hamilton's Equations of Classical Mechanics Lecture I, Aug25, 2014 Newton, Lagrange and Hamilton’s Equations of Classical Mechanics Introduction What this course is about... BOOK Goldstein is a classic Text Book Herbert Goldstein ( June 26, 1922 January 12, 2005) PH D MIT in 1943; Then at Harvard and Columbia first edition of CM book was published in 1950 ( 399 pages, each page is about 3/4 in area compared to new edition) Third edition appeared in 2002 ... But it is an old Text book What we will do different ?? Start with Newton’s Lagrange and Hamilton’s equation one after the other Small Oscillations: Marion, why is it important ??? we start right in the beginning talking about small oscillations, simple limit Phase Space Plots ( Marian, page 159 ) , Touch nonlinear physics and Chaos, Symmetries Order in which Chapters are covered is posted on the course web page Classical Encore 1-4pm, Sept 29, Oct 27, Dec 1 Last week of the Month: Three Body Problem.. ( chaos ), Solitons, may be General Relativity We may not cover scattering and Rigid body dynamics.. the topics that you have covered in Phys303 and I think there is less to gain there.... —————————————————————————— Newton’s Equation, Lagrange and Hamilton’s Equations Beauty Contest: Write Three equations and See which one are the prettiest?? (Simplicity, Mathematical Beauty...) Same Equations disguised in three different forms (I)Lagrange and Hamilton’s equations are scalar equations unlike Newton’s equation.. (II) To apply Newtons’ equation, Forces of constraints are needed to describe constrained motion (III) Symmetries are best described in the Lagrangian formulation (IV)For rectangular coordinates, Newtons’ s Eqn are the easiest.. Try finding components of acceleration in spherical polar coordinates... n spherical coordinates the position of a point is written, In other words, Newton’s equation look different in different ccordinate systems, but Lagrange and Hamilton’s equations are same in all ccordinates The line element for an infinitesimal displacement from (r, θ, ϕ) to (r + dr, θ + dθ, ϕ + dϕ) is dr = dr rˆ + r dθ θˆ + r sin θ dϕ ϕˆ. where 2 rˆ = sin(θ) cos(ϕ)ˆı + sin(θ) sin(ϕ)ˆ+ cos(θ)kˆ θˆ = cos(θ) cos(ϕ)ˆı + cos(θ) sin(ϕ)ˆ− sin(θ)kˆ ϕˆ = − sin(ϕ)ˆı + cos(ϕ)ˆ are the local orthogonal unit vectors in the directions of increasing r, θ, ϕ , respectively, and ˆı, ˆ, kˆ are the unit vectors in Cartesian space. Thus the differential solid angle is dS dΩ = r = sin θ dθ dϕ. (1) r2 The volume element is, dV = r2 sin θ dr dθ dϕ. (2) Position, velocity and acceleration are, r = rˆr v =r ˙ˆr + r θ˙ θˆ + r ϕ˙ sin θϕˆ a = r¨ − r θ˙2 − r ϕ˙ 2 sin2 θ ˆr + r θ¨ + 2r ˙ θ˙ − r ϕ˙ 2 sin θ cos θ θˆ + rϕ¨ sin θ + 2r ˙ ϕ˙ sin θ + 2r θ˙ ϕ˙ cos θ ϕˆ π Check if Equations make sense: (a) Dimension, (b) some limiting case, In the case of a constant φ or θ = 2 , this reduces to vector calculus in polar coordinates. Between Lagrange and Hamilton: Hamilton’s equations reveal symmetry between ”q” and ”p” For covariant Formulation, there are problems in writing L and H in covariant forms if there are forces between particles other than EM force Newton’s equations: Second Order Lagrange Equations: Second Order ( What is L.. For simple mechanical system, for particles in EMF and also for Field Theory.. Two ways to derive these equations: We will follow the second way ( Chapter II ) , using Hamilton’s principle What is Field Theory: Continuous systems and Fields... ... ) Write the Lagrange Equations for Field... Note, for point particles (NR), q’s are mechanical variables and ”t” is a monotonic parameter. But in classical field theory, both q and t are parameters...defining a point in space-time continuum at which Field Variables are determined... Therefore covariant description is natural for fields... Hamilton’s equation: two first order for each independent coordinates As a mathematical problem, transition from L to H mechanics corresponds to changing variables from ( q, q’, t ) to (q, p, t) Pro- cedure for Switching variable can be done using Legendre Transformation..... or you can derive them from Hamilton’s principle... If we want to construct quantum Field Theory of say EMF, we must obtain its description in the language of Lagrangian; That is, we cannot start with Maxwell’s equation and quantize it, just like quantization of any particle, we have to start with its classical Hamiltonian, we cannot start with Newton’s equations... Galileo’s Relativity Principle Equations of motion are invariant under Galilean transformation, r~0 = ~r − V~ t t0 = t 3 Hamilton’s principle Long history.... Minimal principles in physics have long history... First..Second Century BC, Reflection of light 1657, Fermat’s principle, Laws of refraction...., Least Time In physics, Hamilton’s principle is William Rowan Hamilton’s formulation of the principle of stationary action . It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton’s principle also applies to classical fields such as the electromagnetic and gravitational fields, and has even been extended to quantum mechanics, quantum field theory and criticality theories. The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravity. Quantum mechanics and quantum field theory In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all imaginable paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, that gives the probability amplitudes of the various outcomes. Although equivalent in classical mechanics with Newton’s laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynman’s path integral formulation of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell’s equations can be derived as conditions of stationary action. Lagrange Equations for particle in an Electromagnetic Field If velocity dependent forces are such that the force can be written in terms of a scalar function U, d F = Q = −∂ U + ∂ U (3) k k k dt q˙k then we can define L = T − U which will satisfy Lagrange equation. It turns out that the equations are satisfied for all fundamental forces of nature and therefore, there is something very fundament and special about the form of the equations. Home Work Problems ( Brief Review in Class) (1) Chapter I, 22, 23 (2) Find equations of motion for a spherical pendulum of mass m and length b. (3) Consider the motion of a bead along a vertical circle of radius r which rotates with angular velocity ω around the vertical axis passing from the center of the center. Obtain the Lagrangian and equation of motion of the bead. (4) Consider a system of two identical pendulums of unit length and unit mass in a gravitational field g. Suppose the pendulums are connected by a weightless spring of spring constant k. Assuming simple harmonic motion, obtain equations of motion of the system. (5) Show that for a particle of mass m and charge e in an electromagnetic field of scalar potential φ and vector potential A~ is, β maα = −e∂αφ − e∂tAα + eαβγ v γµν ∂µAν (4) 4 where a is acceleration and v is velocity. Use the following identity αβγ γµν = δαµδβν − δαν δβν . (6) Find the Lagrangian and the equations of motion in the following cases of a simple pendulum of mass m whose point of support (a) moves uniformly on a vertical circle with constant frequency γ, (b) oscillates horizontally in the plane of motion of the pendula according to the law x = a cos γt , (c) oscillates vertically according to the law y = a cos γt..
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