Analytical Mechanics. Phys 601
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Rotational Motion (The Dynamics of a Rigid Body)
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates. -
Chapter 5 the Relativistic Point Particle
Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story. -
Branched Hamiltonians and Supersymmetry
Branched Hamiltonians and Supersymmetry Thomas Curtright, University of Miami Wigner 111 seminar, 12 November 2013 Some examples of branched Hamiltonians are explored, as recently advo- cated by Shapere and Wilczek. These are actually cases of switchback poten- tials, albeit in momentum space, as previously analyzed for quasi-Hamiltonian dynamical systems in a classical context. A basic model, with a pair of Hamiltonian branches related by supersymmetry, is considered as an inter- esting illustration, and as stimulation. “It is quite possible ... we may discover that in nature the relation of past and future is so intimate ... that no simple representation of a present may exist.” – R P Feynman Based on work with Cosmas Zachos, Argonne National Laboratory Introduction to the problem In quantum mechanics H = p2 + V (x) (1) is neither more nor less difficult than H = x2 + V (p) (2) by reason of x, p duality, i.e. the Fourier transform: ψ (x) φ (p) ⎫ ⎧ x ⎪ ⎪ +i∂/∂p ⎪ ⇐⇒ ⎪ ⎬⎪ ⎨⎪ i∂/∂x p − ⎪ ⎪ ⎪ ⎪ ⎭⎪ ⎩⎪ This equivalence of (1) and (2) is manifest in the QMPS formalism, as initiated by Wigner (1932), 1 2ipy/ f (x, p)= dy x + y ρ x y e− π | | − 1 = dk p + k ρ p k e2ixk/ π | | − where x and p are on an equal footing, and where even more general H (x, p) can be considered. See CZ to follow, and other talks at this conference. Or even better, in addition to the excellent books cited at the conclusion of Professor Schleich’s talk yesterday morning, please see our new book on the subject ... Even in classical Hamiltonian mechanics, (1) and (2) are equivalent under a classical canonical transformation on phase space: (x, p) (p, x) ⇐⇒ − But upon transitioning to Lagrangian mechanics, the equivalence between the two theories becomes obscure. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
SPEED MANAGEMENT ACTION PLAN Implementation Steps
SPEED MANAGEMENT ACTION PLAN Implementation Steps Put Your Speed Management Plan into Action You did it! You recognized that speeding is a significant Figuring out how to implement a speed management plan safety problem in your jurisdiction, and you put a lot of time can be daunting, so the Federal Highway Administration and effort into developing a Speed Management Action (FHWA) has developed a set of steps that agency staff can Plan that holds great promise for reducing speeding-related adopt and tailor to get the ball rolling—but not speeding! crashes and fatalities. So…what’s next? Agencies can use these proven methods to jump start plan implementation and achieve success in reducing speed- related crashes. Involve Identify a Stakeholders Champion Prioritize Strategies for Set Goals, Track Implementation Market the Develop a Speed Progress, Plan Management Team Evaluate, and Celebrate Success Identify Strategy Leads INVOLVE STAKEHOLDERS In order for the plan to be successful, support and buy-in is • Outreach specialists. needed from every area of transportation: engineering (Federal, • Governor’s Highway Safety Office representatives. State, local, and Metropolitan Planning Organizations (MPO), • National Highway Transportation Safety Administration (NHTSA) Regional Office representatives. enforcement, education, and emergency medical services • Local/MPO/State Department of Transportation (DOT) (EMS). Notify and engage stakeholders who were instrumental representatives. in developing the plan and identify gaps in support. Potential • Police/enforcement representatives. stakeholders may include: • FHWA Division Safety Engineers. • Behavioral and infrastructure funding source representatives. • Agency Traffic Operations Engineers. • Judicial representatives. • Agency Safety Engineers. • EMS providers. • Agency Pedestrian/Bicycle Coordinators. • Educators. • Agency Pavement Design Engineers. -
The Lagrangian and Hamiltonian Mechanical Systems
THE LAGRANGIAN AND HAMILTONIAN MECHANICAL SYSTEMS ALEXANDER TOLISH Abstract. Newton's Laws of Motion, which equate forces with the time- rates of change of momenta, are a convenient way to describe mechanical systems in Euclidean spaces with cartesian coordinates. Unfortunately, the physical world is rarely so cooperative|physicists often explore systems that are neither Euclidean nor cartesian. Different mechanical formalisms, like the Lagrangian and Hamiltonian systems, may be more effective at describing such phenomena, as they are geometric rather than analytic processes. In this paper, I shall construct Lagrangian and Hamiltonian mechanics, prove their equivalence to Newtonian mechanics, and provide examples of both non- Newtonian systems in action. Contents 1. The Calculus of Variations 1 2. Manifold Geometry 3 3. Lagrangian Mechanics 4 4. Two Electric Pendula 4 5. Differential Forms and Symplectic Geometry 6 6. Hamiltonian Mechanics 7 7. The Double Planar Pendulum 9 Acknowledgments 11 References 11 1. The Calculus of Variations Lagrangian mechanics applies physics not only to particles, but to the trajectories of particles. We must therefore study how curves behave under small disturbances or variations. Definition 1.1. Let V be a Banach space. A curve is a continuous map : [t0; t1] ! V: A variation on the curve is some function h of t that creates a new curve +h.A functional is a function from the space of curves to the real numbers. p Example 1.2. Φ( ) = R t1 1 +x _ 2dt, wherex _ = d , is a functional. It expresses t0 dt the length of curve between t0 and t1. Definition 1.3. -
Newton's Third Law (Lecture 7) Example the Bouncing Ball You Can Move
3rd Law Newton’s third law (lecture 7) • If object A exerts a force on object B, then object B exerts an equal force on object A For every action there is an in the opposite direction. equal and opposite reaction. B discuss collisions, impulse, A momentum and how airbags work B Æ A A Æ B Example The bouncing ball • What keeps the box on the table if gravity • Why does the ball is pulling it down? bounce? • The table exerts an • It exerts a downward equal and opposite force on ground force upward that • the ground exerts an balances the weight upward force on it of the box that makes it bounce • If the table was flimsy or the box really heavy, it would fall! Action/reaction forces always You can move the earth! act on different objects • The earth exerts a force on you • you exert an equal force on the earth • The resulting accelerations are • A man tries to get the donkey to pull the cart but not the same the donkey has the following argument: •F = - F on earth on you • Why should I even try? No matter how hard I •MEaE = myou ayou pull on the cart, the cart always pulls back with an equal force, so I can never move it. 1 Friction is essential to movement You can’t walk without friction The tires push back on the road and the road pushes the tires forward. If the road is icy, the friction force You push on backward on the ground and the between the tires and road is reduced. -
Newton's Third Law of Motion
ENERGY FUNDAMENTALS – LESSON PLAN 1.4 Newton’s Third Law of Motion This lesson is designed for 3rd – 5th grade students in a variety of school settings Public School (public, private, STEM schools, and home schools) in the seven states served by local System Teaching power companies and the Tennessee Valley Authority. Community groups (Scouts, 4- Standards Covered H, after school programs, and others) are encouraged to use it as well. This is one State lesson from a three-part series designed to give students an age-appropriate, Science Standards informed view of energy. As their understanding of energy grows, it will enable them to • AL 3.PS.4 3rd make informed decisions as good citizens or civic leaders. • AL 4.PS.4 4th rd • GA S3P1 3 • GA S3CS7 3rd This lesson plan is suitable for all types of educational settings. Each lesson can be rd adapted to meet a variety of class sizes, student skill levels, and time requirements. • GA S4P3 3 • KY PS.1 3rd rd • KY PS.2 3 Setting Lesson Plan Selections Recommended for Use • KY 3.PS2.1 3rd th Smaller class size, • The “Modeling” Section contains teaching content. • MS GLE 9.a 4 • NC 3.P.1.1 3rd higher student ability, • While in class, students can do “Guided Practice,” complete the • TN GLE 0307.10.1 3rd and /or longer class “Recommended Item(s)” and any additional guided practice items the teacher • TN GLE 0307.10.2 3rd length might select from “Other Resources.” • TN SPI 0307.11.1 3rd • NOTE: Some lesson plans do and some do not contain “Other Resources.” • TN SPI 0307.11.2 3rd • At home or on their own in class, students can do “Independent Practice,” • TN GLE 0407.10.1 4th th complete the “Recommended Item(s)” and any additional independent practice • TN GLE 0507.10.2 5 items the teacher selects from “Other Resources” (if provided in the plan). -
Analytical Mechanics
A Guided Tour of Analytical Mechanics with animations in MAPLE Rouben Rostamian Department of Mathematics and Statistics UMBC [email protected] December 2, 2018 ii Contents Preface vii 1 An introduction through examples 1 1.1 ThesimplependulumàlaNewton ...................... 1 1.2 ThesimplependulumàlaEuler ....................... 3 1.3 ThesimplependulumàlaLagrange.. .. .. ... .. .. ... .. .. .. 3 1.4 Thedoublependulum .............................. 4 Exercises .......................................... .. 6 2 Work and potential energy 9 Exercises .......................................... .. 12 3 A single particle in a conservative force field 13 3.1 The principle of conservation of energy . ..... 13 3.2 Thescalarcase ................................... 14 3.3 Stability....................................... 16 3.4 Thephaseportraitofasimplependulum . ... 16 Exercises .......................................... .. 17 4 TheKapitsa pendulum 19 4.1 Theinvertedpendulum ............................. 19 4.2 Averaging out the fast oscillations . ...... 19 4.3 Stabilityanalysis ............................... ... 22 Exercises .......................................... .. 23 5 Lagrangian mechanics 25 5.1 Newtonianmechanics .............................. 25 5.2 Holonomicconstraints............................ .. 26 5.3 Generalizedcoordinates .......................... ... 29 5.4 Virtual displacements, virtual work, and generalized force....... 30 5.5 External versus reaction forces . ..... 32 5.6 The equations of motion for a holonomic system . ... -
Climate Solutions Acceleration Fund Winners Announced
FOR IMMEDIATE RELEASE Contact: Amanda Belles, Communications & Marketing Manager (617) 221-9671; [email protected] Climate Solutions Acceleration Fund Winners Announced Boston, Massachusetts (June 3, 2020) - Today, Second Nature - a Boston-based NGO who accelerates climate action in, and through, higher education - announced the colleges and universities that were awarded grant funding through the Second Nature Climate Solutions Acceleration Fund (the Acceleration Fund). The opportunity to apply for funding was first announced at the 2020 Higher Education Climate Leadership Summit. The Acceleration Fund is dedicated to supporting innovative cross-sector climate action activities driven by colleges and universities. Second Nature created the Acceleration Fund with generous support from Bloomberg Philanthropies, as part of a larger project to accelerate higher education’s leadership in cross-sector, place-based climate action. “Local leaders are at the forefront of climate action because they see the vast benefits to their communities, from cutting energy bills to protecting public health,” said Antha Williams, head of environmental programs at Bloomberg Philanthropies. “It’s fantastic to see these forward-thinking colleges and universities advance their bold climate solutions, ensuring continued progress in our fight against the climate crisis.” The institutions who were awarded funding are (additional information about each is further below): Agnes Scott College (Decatur, GA) Bard College (Annandale-on-Hudson, -
Lagrangian Mechanics
Alain J. Brizard Saint Michael's College Lagrangian Mechanics 1Principle of Least Action The con¯guration of a mechanical system evolving in an n-dimensional space, with co- ordinates x =(x1;x2; :::; xn), may be described in terms of generalized coordinates q = (q1;q2;:::; qk)inak-dimensional con¯guration space, with k<n. The Principle of Least Action (also known as Hamilton's principle) is expressed in terms of a function L(q; q_ ; t)knownastheLagrangian,whichappears in the action integral Z tf A[q]= L(q; q_ ; t) dt; (1) ti where the action integral is a functional of the vector function q(t), which provides a path from the initial point qi = q(ti)tothe¯nal point qf = q(tf). The variational principle ¯ " à !# ¯ Z d ¯ tf @L d @L ¯ j 0=±A[q]= A[q + ²±q]¯ = ±q j ¡ j dt; d² ²=0 ti @q dt @q_ where the variation ±q is assumed to vanish at the integration boundaries (±qi =0=±qf), yields the Euler-Lagrange equation for the generalized coordinate qj (j =1; :::; k) à ! d @L @L = ; (2) dt @q_j @qj The Lagrangian also satis¯esthesecond Euler equation à ! d @L @L L ¡ q_j = ; (3) dt @q_j @t and thus for time-independent Lagrangian systems (@L=@t =0)we¯nd that L¡q_j @L=@q_j is a conserved quantity whose interpretationwill be discussed shortly. The form of the Lagrangian function L(r; r_; t)isdictated by our requirement that Newton's Second Law m Är = ¡rU(r;t)describing the motion of a particle of mass m in a nonuniform (possibly time-dependent) potential U(r;t)bewritten in the Euler-Lagrange form (2). -
Lagrangian Mechanics - Wikipedia, the Free Encyclopedia Page 1 of 11
Lagrangian mechanics - Wikipedia, the free encyclopedia Page 1 of 11 Lagrangian mechanics From Wikipedia, the free encyclopedia Lagrangian mechanics is a re-formulation of classical mechanics that combines Classical mechanics conservation of momentum with conservation of energy. It was introduced by the French mathematician Joseph-Louis Lagrange in 1788. Newton's Second Law In Lagrangian mechanics, the trajectory of a system of particles is derived by solving History of classical mechanics · the Lagrange equations in one of two forms, either the Lagrange equations of the Timeline of classical mechanics [1] first kind , which treat constraints explicitly as extra equations, often using Branches [2][3] Lagrange multipliers; or the Lagrange equations of the second kind , which Statics · Dynamics / Kinetics · Kinematics · [1] incorporate the constraints directly by judicious choice of generalized coordinates. Applied mechanics · Celestial mechanics · [4] The fundamental lemma of the calculus of variations shows that solving the Continuum mechanics · Lagrange equations is equivalent to finding the path for which the action functional is Statistical mechanics stationary, a quantity that is the integral of the Lagrangian over time. Formulations The use of generalized coordinates may considerably simplify a system's analysis. Newtonian mechanics (Vectorial For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using Newtonian mechanics) mechanics would require solving for the time-varying constraint force required to Analytical mechanics: keep the bead in the groove. For the same problem using Lagrangian mechanics, one Lagrangian mechanics looks at the path of the groove and chooses a set of independent generalized Hamiltonian mechanics coordinates that completely characterize the possible motion of the bead.