LECTURE 1 PHY5521 Classical Mechanics Honour to Masters Level
- 2. 1PHY-2 : CLASSICAL MECHANICS : Syllabus:
- 3. UNIT-III : Cyclic coordinates, Integrals of the motion, Concepts of symmetry, homogeneity and isotropy, Invariance under Galilean transformations Hamilton’s equation of motion: Legendre’s dual transformation, Principle of least action; derivation of equations of motion; variation and end points; Hamilton’s principle and characteristic functions; Hamilton- Jacobi equation. UNIT-IV : Central force: Definition and properties, Two-body central force problem, closure and stability of circular orbits; general analysis of orbits; Kepler’s laws and equation, Classification of orbits, differential equation of orbit, Virial Theorem. UNIT-V : Canonical transformation; generating functions; Properties; group property; examples; infinitesimal generators; Poisson bracket; Poisson theorems; angular momentum PBs; Transition from discrete to continuous system, small oscillations (longitudinal oscillations in elastic rod); normal modes and coordinates.
- 4. Introduction to Classical Mechanics Introduction to Newtonian Mechanics Classical mechanics deals with the motion of physical bodies at the macroscopic level. Galileo and Sir Isaac Newton laid its foundation in the 17th century. As Newton’s laws of motion provide the basis of classical mechanics, it is often referred to as Newtonian mechanics. There are two parts in mechanics: kinematics and dynamics. Kinematics deals with the geometrical description of the motion of objects without considering the forces producing the motion. Dynamics is the part that concerns the forces that produce changes in motion or the changes in other properties. This leads us to the concept of force, mass and the laws that govern the motion of objects. To apply the laws to different situations, Newtonian mechanics has since been reformulated in a few different forms, such as the Lagrange, the Hamilton and the
- 5. Mechanics is a branch of Physics which deals with physical objects in motion and at rest under the influence of external and internal interactions.
- 7. The mathematical study of the motion of everyday objects and the forces that affect them is called classical mechanics. Laws of motion formulated by Galileo, Newton, Lagrange, Hamilton, Maxwell which comes before quantum theory are referred to as Classical Mechanics.
- 8. Failure of classical mechanics • Classical mechanics explains correctly the motion of celestial bodies like planets, stars etc. macroscopic objects but it is failed to describe the behaviour of microscopic objects such as atoms, nuclei, and elementary particles.
- 9. • According to classical mechanics, if we consider the case of an electron moving around the nucleus, its energy should decrease because a charged particle losses energy in the form of electromagnetic energy and therefore its velocity should decrease continuously. The result is that electron comes closer to the nucleus until it collapses. This shows the instability of the atom. • It is in contradiction to the observed fact of the stability of atom. The classical mechanics fails to explain the stability of atom.
- 10. FRAMES OF REFERENCE The most basic concepts for the study of motion are space and time, both of which are assumed to be continuous. To describe the motion of a body, one has to specify its position in space as a function of time. To do this, a co-ordinate system is used as a frame of reference. One convenient co-ordinate system we frequently use is the cartesian or rectangular co- ordinate system. Cartesian Co-ordinates (x, y, z) The position of a point P in a cartesian co-ordinate system, as shown in Fig. 1.1(a), is specified by three co-ordinates (x, y, z) or (x1, x2, x3) or by the position vector r. A vector quantity will be denoted by boldface type (as r), while the magnitude will be represented by the symbol itself (as 𝒓 =r/r). A unit vector in the direction of the vector r is denoted by the corresponding letter with a circumflex over it (as ). In terms of the co-ordinates, the vector and the magnitude of the vector is given by; 𝒓 = 𝒊𝒙 + 𝒋𝒚 + 𝒌𝒛
- 11. Elementary lengths in the direction of x, y, z: dx, dy, dz Elementary volume: dx dy dx Cartesian co-ordinates are convenient in describing the motion of objects in a straight line. However, in certain problems, it is convenient to use nonrectangular co-ordinates.
- 12. 𝐜𝐨𝐬 𝜽 = 𝒙 𝒓 𝐬𝐢𝐧 𝜽 = 𝒚 𝒓 𝐭𝐚𝐧 𝜽 = 𝒚 𝒙
- 13. NEWTON’S LAWS OF MOTION Newton’s First Law of Motion (Law of inertia) Every object continues in its state of rest or uniform motion in a straight line unless a net external force acts on it to change that state. Newton's First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. Isaac Newton (1642–1727)
- 14. Newton’s first law indicates that the state of a body at rest (zero velocity) and a state of uniform velocity are completely equivalent. No external force is needed in order to maintain the uniform motion of a body; it continues without change due to an intrinsic property of the body that we call inertia. Because of this property, the first law is often referred to as the law of inertia. Inertia is the natural tendency of a body to remain at rest or in uniform motion along a straight line. Quantitatively, the inertia of a body is measured by its mass. Newton made the first law more precise by introducing definitions of quantity of motion and amount of matter which we now call momentum and mass respectively. The momentum of a body is simply proportional to its velocity. The coefficient of proportionality is a constant for any given body and is called its mass. Denoting mass by m and momentum vector by p; p = mv ………(1.6) where v is the velocity of the body. Mathematically, Newton’s first law can be expressed in the following way. In the absence of an external force acting on a body p = mv = constant………(1.7) This is the law of conservation of momentum.
- 16. Newton’s Second Law of Motion (Law of force) The rate of change of momentum of an object is directly proportional to the force applied and takes place in the direction of the force. ∴ 𝑭 = 𝒅𝒑 𝒅𝒕 ∴ 𝑭 = 𝒎 𝒅𝒗 𝒅𝒕 = 𝒎𝒂 ∴ 𝑭 = 𝒎 𝒅𝟐 𝒓(𝒕) 𝒅𝒕𝟐
- 18. Newton’s Third Law of Motion (Law of action and reaction) Whenever a body exerts a force on a second body, the second exerts an equal and opposite force on the first.
- 20. INERTIAL AND NON-INERTIAL FRAMES
- 21. Newton’s first law does not hold in every reference frame. When two bodies fall side by side, each of them is at rest with respect to the other while at the same time it is subject to the force of gravity. Such cases would contradict the stated first law. Reference frames in which Newton’s law of inertia holds good are called inertial reference frames. The remaining laws are also valid in inertial reference frames only. The acceleration of an inertial reference frame is zero and therefore it moves with a constant velocity. Any reference frame that moves with constant velocity relative to an inertial frame is also an inertial frame of reference. For simple applications in the laboratory, reference frames fixed on the earth are inertial frames. For astronomical applications, the terrestrial frame cannot be regarded as an inertial frame. A reference frame where the law of inertia does not hold is called a non-inertial reference frame.
- 22. Inertial frame of reference Non-inertial frame of reference The body does not accelerate. The body undergoes acceleration In this frame, a force acting on a body is a real force. The acceleration of the frame gives rise to a pseudo force. A Pseudo force (also called a fictitious force, inertial force or d'Alembert force) is an apparent force that acts on all masses whose motion is described using a non-inertial frame of reference frame, such as rotating reference frame.
- 23. INERTIAL FRAMES
- 27. Mechanics of a particle: Conservation laws Conservation of Linear momentum
- 28. Conservation of Angular momentum
- 30. Conservation of Energy Kinetic Energy and Work-Energy theorem 𝑭 = 𝒎 𝒅𝒗 𝒅𝒕 Hence, “The work-energy theorem states that the net work done by the forces on an object equals the change in its kinetic energy.” According to Newtons second law;
- 36. Mechanics of a system of particles External and internal forces
- 38. Centre of mass
- 44. Principal of Virtual work The constraint forces do work in dissipative and rheonomic constraints. These constraint forces remain largely undetermined. Hence, it is useful if we could plan to eliminate these forces from the equations. This is done by the use of virtual work through virtual displacements of the system. A virtual displacement is any imaginary displacement of the system which is consistent with the constraint relation at a given instant. Principle of virtual work states that the work done is zero in the case of an arbitrary virtual displacement of a system from a position of equilibrium. Mechanical energy is not conserved during the constrained motion, the constraint is called dissipative A mechanical system is rheonomous if its equations of constraints contain the time as an explicit variable.
- 45. Let the total force on the system of ith particle can be expressed as; 𝑭𝒊 = 𝑭𝒊 𝒂 + 𝒇𝒊 (𝟏) 𝑭𝒊 𝒂 is the applied force and 𝒇𝒊 is the force of constraint. The virtual work of the force Fi in the virtual displacement 𝜹𝒓𝒊 will be zero. 𝜹𝑾 = 𝒊=𝟏 𝑵 𝑭𝒊𝜹𝒓𝒊 = 𝟎 (𝟐) 𝜹𝑾 = 𝒊=𝟏 𝑵 𝑭𝒊 𝒂 𝜹𝒓𝒊 + 𝒊=𝟏 𝑵 𝒇𝒊 𝜹𝒓𝒊 = 𝟎 (𝟐) The virtual work of the forces of constraint is zero. ∴ 𝜹𝑾 = 𝒊=𝟏 𝑵 𝑭𝒊 𝒂 𝜹𝒓𝒊 = 𝟎 i.e. for equilibrium of a system, the virtual work of the applied forces is zero.
- 46. D’Alembert’s principle: The principle states that the sum of the differences between the forces acting on a system and the time derivatives of the momenta of the system itself along any virtual displacement consistent with the constraint of the system is zero. According to Newtons second law, the force acting on the ith particle is given by; 𝑭𝒊 = 𝒅𝒑𝒊 𝒅𝒕 = 𝒑𝒊 ∴ 𝑭𝒊 −𝒑𝒊 = 𝟎 These equations mean that any particle in the system is in equilibrium under actual force Fi and reversed force 𝒑𝒊.
- 47. Therefore, leads to virtual displacement 𝜹𝒓𝒊; 𝒊=𝟏 𝑵 (𝑭𝒊−𝒑𝒊) 𝜹𝒓𝒊 = 𝟎 But, 𝑭𝒊 = 𝑭𝒊 𝒂 + 𝒇𝒊 then, 𝒊=𝟏 𝑵 (𝑭𝒊 𝒂 −𝒑𝒊) 𝜹𝒓𝒊 + 𝒊=𝟏 𝑵 𝒇𝒊𝜹𝒓𝒊 = 𝟎 Since, virtual work of constraints is zero. 𝒊=𝟏 𝑵 (𝑭𝒊 𝒂 −𝒑𝒊) 𝜹𝒓𝒊 = 𝟎 This is known as D’Alembert’s principle.
- 48. Constraints Motion along a specified path is the simplest example of constrained motion.
- 50. •If constraint relations are in the form of f (r1, r2, r3…rn, t)=0 •If constraint relations are or can be made independent of velocity •Ex- a cylinder rolling without sliding down an inclined plane. Holonomic • If constraint relations are not holonomic i.e. these relations are irreducible functions of velocities. If system is represented in the form of inequalities r>a • Ex. A sphere rolling without sliding down an inclined plane. Non-holonomic •If constraint relations do not depend explicitly on time. •Ex- Rigid body Scleronomic •If constraint relations depend explicitly on time •Ex- A bead sliding on a moving wire. Rheonomic •If constraint equations are in the form of equations. Bilateral • If the constraint relations are expressed in form of inequalities. Unilateral •If forces of constraint do not do any work and total mechanical energy of the system is conserved while performing constrained motion. •Ex- Simple pendulum with rigid support Conservative •If the forces of constraint do work and the total mechanical energy is not conserved. •Ex- Pendulum with variable length. Dissipative Classification of constraints:
- 51. Degrees of freedom The minimum number of independent variables or coordinates required to specify the position of a dynamical system, consisting of one or more particles, is called the number of degrees of freedom of the system.
- 55. Rigid Body The motion of a rigid body is a constrained motion as distance between any two points remains constant. 𝒓𝒊 − 𝒓𝒋 = 𝑪𝒊𝒋 𝒓𝒊 − 𝒓𝒋 𝟐 = 𝑪𝒊𝒋 𝟐 𝒓𝒊 − 𝒓𝒋 𝟐 − 𝑪𝒊𝒋 𝟐 = 𝟎
- 58. x = 0 x = L
- 59. Examples of Holonomic and Non-holonomic constraints:
- 67. Motion of a body on an inclined plane under gravity:
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