PHL511 Classical Mechanics Syllabus PDF

Course Name: PHL511 – Classical Mechanics Syllabus: Revision of Newtonian mechanics, constraints, Generalized coordinates Lagrange’s equations of motion, Noethers theorem. Hamilton’s function and Hamilton’s equation of motion, Legendre transform, Phase space, Phase trajectories, Principle of least a...

Course Name: PHL511 – Classical Mechanics Syllabus: Revision of Newtonian mechanics, constraints, Generalized coordinates Lagrange’s equations of motion, Noethers theorem. Hamilton’s function and Hamilton’s equation of motion, Legendre transform, Phase space, Phase trajectories, Principle of least action, Hamiltonian principle. Two body central force problem, Kepler problem, Scattering, Virial theorem. Non-inertial frames of reference and pseudo forces, Elements of rigid body dynamics. Small oscillations, Normal mode analysis, Normal modes of a harmonic chain. Principle and postulate of relativity, Lorentz transformation , Length contraction, Time dilation and the Doppler Effect, Relativistic invariance of physical laws.  Cartesian Co-ordinates (x, y, z) Three co-ordinates (x, y, z) Unit vector in the direction of the vector r is denoted by the corresponding letter with P In terms of the co-ordinates, the vector and the magnitude of the vector are given by + Where are unit vectors along the rectangular axes x, y and z Elementary lengths in the direction of x, y, z: dx, dy, dz Elementary volume: dx dy dx The position of point P can be described by the co-ordinates (r, q) called plane polar co-ordinates. The rectangular co-ordinates of P are (x, y). Spherical Polar Co-ordinates (r, , ) OQ = r sin  OC = PQ = r cos   Newton’s First Law of Motion 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. 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 p = mv In the absence of an external force acting on a body p = mv = constant This is the law of conservation of momentum  Newton’s Second Law of Motion 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.  Newton’s Third Law of Motion Whenever a body exerts a force on a second body, the second exerts an equal and opposite force on the first Every action there is an equal and opposite reaction.  INERTIAL AND NON-INERTIAL FRAMES 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. A reference frame where the law of inertia does not hold is called a non-inertial reference frame.  Conservation of Linear Momentum If the total force acting on a particle is zero, then the linear momentum p is conserved  Angular Momentum and Torque Angular momentum and torque are two important quantities in rotational motion. A force causes linear acceleration whereas a torque causes angular acceleration. The angular momentum of a particle about a point O (say origin), denoted by L, is defined as L=rXp where r is the radius vector of the particle The torque (N) or moment of a force about O is which is perpendicular to the plane containing the vectors r and F points in the direction of the advance of a right hand screw from r to F which is the analogue of Newton’s second law in rotational motion Conservation of Angular Momentum If the torque N acting on a particle is zero, the angular momentum L is a constant. Planets moving around the sun and satellites around the earth are some of the very common examples  Work Done by a Force Work done by an external force in moving a particle from position 1 to Position 2 where T2 and T1 are the kinetic energies of the particle in positions 2 and 1 respectively. If T2 > T1, W12 > 0, work is done by the force on the particle and as a result the kinetic energy of the particle is increased. If T1 > T2, W12 < 0, work is done by the particle against the force and as a result the kinetic energy of the particle is decreased.  Conservative Force If the force acting on a system is such that the work done along a closed path is zero -ve sign indicated that F in in direction of decreasing V  Conservation of Energy The work done by a force F in moving a particle of mass m from position 1 to position 2. Now consider the work done W12 by taking F to be a conservative force derivable from a potential V. Then W12 takes the form energy conservation theorem If the force acting on a particle is conservative, then the total energy of the particle, T + V, is a constant. Particle of mass m moves under a force F = – cx3, where c is a positive constant. (i) Find the potential energy function; (ii) If the particle starts from rest at x = – a, what is its velocity when it reaches x = 0? (iii) Where in the subsequent motion does it come to rest? A disc of mass m and radius r rolls down an inclined plane of angle . Find the acceleration of the disc and the frictional force Forces acting on the disc are the weight mg, the reaction R and the frictional force f. Let the acceleration of the disc be a The unbalanced force on the disc = mg sin – f This must be equal to ma Hence, mg sin – f Moment of inertia of the disc about its point of contact Equating the two expressions for torque  System of Particles CENTRE OF MASS The mass of a point particle is concentrated at a particular point. Consider the motion of a system of n particles, a point which behaves as if the entire mass of the system is concentrated at that point. This point is called the centre of mass of the system. The centre of mass C of a system of particles whose radius vector is R is related to the masses mi and radius vectors ri of all n particles of the system by the equation M -total mass of the system.  A frame of reference with the centre of mass as the origin is called the centre of mass frame of reference. In this frame of reference, obviously, the position vector of the centre of mass R is equal to zero. Consequently, the linear momentum P of the system (dR/dt) is also zero. For a continuous body, the coordinates of the centre of mass are  CONSERVATION OF LINEAR MOMENTUM Consider a system of n particles of masses m1, m2, m3,... mn. Let their position vectors at time t be r1, r2, r3,... rn. The force acting on the ith particle Fi has two parts: (i) a force applied on the system from outside or external force (ii) an internal force which is a force among the particles of the system.  CONSERVATION OF LINEAR MOMENTUM Consider a system of n particles of masses m1, m2, m3,... mn. Let their position vectors at time t be r1, r2, r3,... rn. The force acting on the ith particle Fi has two parts: (i) a force applied on the system from outside or external force (ii) an internal force which is a force among the particles of the system. Newton’s second law for the ith particle of the system can be written as Since Fii = 0, where Fie is the external force on the ith particle and Fij is the internal force on the ith particle due to the jth one Assuming that Newton’s third =0 law is valid for the internal force The total external force acting on the system The sum = total linear momentum of the system the law of conservation of linear momentum of a system of particles: If the external force acting on a system of particles is zero, then the total linear momentum of the system is conserved. When external force acting on a system is zero, it is called a closed system. For a closed system, linear momentum is conserved. Relation connecting the total linear momentum and the velocity of the centre of mass The centre of mass moves as if the total external force were acting on the entire mass of the system concentrated at the centre of mass.  ANGULAR MOMENTUM The angular momentum L of a system of particles which is defined The position vector of the centre of mass of the system and that of the ith particle  L vanishes as it defines the radius vector of the centre of mass in the co-ordinate system in which the origin is the centre of mass. The quantity where VCM is the velocity of the centre of mass w.r.t. the origin O. The total angular momentum about a point O is equal to the sum of the angular momentum of the system concentrated at the centre of mass and the angular momentum of the system of particles about the centre of mass  CONSERVATION OF ANGULAR MOMENTUM Consider the angular momentum of a system of n particles which is defined as The first term on the right is zero since the vector product of a vector with itself is zero which is zero if the internal forces are central, that is, the internal forces are along the line joining the two particles. is the torque due to the external force on the ith particle Where is the total external torque acting on the system If the total torque due to external forces on a system of particles is zero, then the total angular momentum is a constant of motion.  KINETIC ENERGY FOR A SYSTEM OF PARTICLES For a system of particles the kinetic energy of the system The position of the centre of mass of the system and that of the ith particle is  vanishes as it defines the radius vector of the =0 centre of mass in the co-ordinate system in which the origin is the centre of mass Thus, like angular momentum, the kinetic energy also consists of two parts: (i) the kinetic energy obtained if all the mass were concentrated at the centre of mass, (ii) the kinetic energy of motion about the centre of mass.  ENERGY CONSERVATION OF A SYSTEM OF PARTICLES The energy conservation law of a single particle system can easily be extended to a system of particles. The force acting on the ith particle is given by where T is the total kinetic energy of the system If both Fie and Fij are conservative, they are derivable from potential functions where the subscript i on the del operator indicates that the derivative is with respect to the co-ordinates of the ith particle where the factor ½ is introduced to avoid each member of a pair being included twice, first in the i summation and then in the j summation. As the internal and external forces are derivable from potentials, it is possible to define a total potential energy V of the system: which gives the energy conservation law: For a conservative system of n particles, the total energy E = T + V is constant Masses of 1, 2 and 3 kg are located at positions respectively. If their velocities are find the position and velocity of the centre of mass. Also, find the angular momentum of the system with respect to the origin Radius vector of the centre of mass Lagrangian Formulation  CONSTRAINTS Constrained motion: A motion that cannot proceed arbitrarily in any manner Holonomic Constraints Expressible as equations connecting the co-ordinates and time i) In a rigid body, the distance between any two particles of the body remains constant during motion where cij is the distance between the particles i and j at ri and rj. (ii) The sliding of a bead on a circular wire of radius a in the xy-plane Holonomic constraints are also known as integrable constraints  Non-holonomic Constraints Non-holonomic constraints are those which are not expressible In non-holonomic constraints, if the constraints are expressible as relations among the velocities of the particles of the system If these equations of non-holonomic constraints can be integrated to give relations among the co-ordinates, then it become holonomic. (i) The constraint involved in the example of a particle placed on the surface of a sphere is non-holonomic, expressed as the inequality where a is the radius of the sphere (ii) Gas molecules in a spherical container of radius R. If ri is the position vector of the ith molecule Here, the centre of the sphere is the origin of the co-ordinate system.  Scleronomous Constraints Scleronomous constraint is one that is independent of time A pendulum with an inextensible string of length l0 is described by the equation As the constraint equation is independent of time, it is a scleronomous constraint Rheonomous Constraints Rheonomous constraint contains time explicitly (dependent of time) where l(t) is the length of the string at time t.  GENERALIZED CO-ORDINATES Degrees of Freedom Number of independent ways in which a mechanical system can move without violating any constraint is called the number of degrees of freedom of the system It is the minimum possible number of co-ordinates required to describe the system completely When a particle moves in space, it has three degrees of freedom. If it is constrained to move along a space curve it has only one degree of freedom It has two degrees of freedom if it moves in a plane  Generalized Co-ordinates For a system of N particles, free from constraints, we require a total of 3N independent co-ordinates to describe its configuration completely Let there are k constraints of the type acting on the system. Now the system has only 3N – k independent coordinates or degrees of freedom. These 3N – k independent co-ordinates represented by the variables are called the generalized co-ordinates. In terms of the new co- ordinates, the old co-ordinates r1, r2,..., rN can be written as These are the transformation equations from the set of variables r1 to q1 variables. In analogy with cartesian co-ordinates, -defined as generalized velocities time derivatives  Configuration Space Configuration of a system can be specified completely by the values of n = 3N – k independent generalized co-ordinates q1, q2,...,qn. It is convenient to think of the n q’s as the co-ordinates of a point in an n-dimensional space. This n-dimensional space is called the configuration space with each dimension represented by a co-ordinate. As the generalized co-ordinates are not necessarily position co- ordinates, configuration space is not necessarily connected to the physical 3-dimensional space and the path of motion also does not necessarily resemble the path in space of actual particle.  PRINCIPLE OF VIRTUAL WORK A virtual displacement, denoted by dri, refers to an imagined, infinitesimal, instantaneous displacement of the co-ordinate that is consistent with the constraints It is called virtual as the displacement is instantaneous. As there is no actual motion of the system, the work done by the forces of constraint in such a virtual displacement is zero. It is different from an actual displacement dri of the system occurring in a time interval dt. Consider a scleronomic system of N particles in equilibrium. Let Fi be the force acting on the ith particle. The force Fi is a vector addition of the externally applied force and the forces of constraints fi.  If dri is a virtual displacement of the ith particle, the virtual work done dWi on the ith particle is given by If the system is in equilibrium, the total force on each particle must be zero: Fi = 0 for all i. Therefore, the dot product is also zero The total virtual work done on the system dW is the sum of the above vanishing products Under a virtual displacement, the work done by the forces of constraints is zero. This is valid for rigid bodies and most of the constraints that commonly occur. which is the principle of virtual work In an N-particle system, the total work done by the external forces when virtual displacements are made is called virtual work and the total virtual work done is zero- principle of virtual work deals only with statics.  D’ALEMBERT’S PRINCIPLE The principle of virtual work deals only with statics and the general motion of the system is not relevant here A principle that involves the general motion of the system was suggested by D’ Alembert Consider the motion of an N-particle system. Let the force acting on the ith particle be Fi. By Newton’s law This means that the ith particle in the system will be in equilibrium under a force equal to the actual force plus a “reversed effective force”, as named by D’Alembert. Then dynamics reduces to statics = a virtual displacement Restricting to situations where the virtual work done by forces of constraints is zero -D’Alembert’s Principle.  LAGRANGE’S EQUATIONS Lagrange used D’Alembert’s principle as the starting point to derive the equations of motion The virtual displacements in Eq. (3.20) are not independent Consider a system with N particles at r1, r2,..., rN having k equations of holonomic constraints. The system will have n = 3N – k generalized coordinates q1, q2,...,qn. The transformation equations from the r variables to the q variables are given by dqj’s are the virtual displacements of generalized co-ordinates Quantity Qj is the jth component of the generalized force Q. The generalized force components need not have the dimension of force as the q’s need not have the dimension of length However, Q jdqj must have the dimension of work. where T is the total kinetic energy of the system The dq’s are independent and therefore each of the coefficients must separately vanish. From which it follows that if the external forces Fi are conservative: where V = V (r1, r2,..., rN). If potential V is function of position only then These n equations, one for each independent generalized co- ordinate, are known as Lagrange’s equations  LAGRANGE’S EQUATIONS-Non Conservative System In certain systems the forces acting are not conservative, say where a part is derivable from a potential and the other is dissipative. In such cases, Lagrange’s equations can be written as where L contains the potential of the conservative forces represents the force not arising from that potential. represents the force not arising from that potential  KINETIC ENERGY IN GENERALIZED COORDINATES Kinetic energy of a particle of mass m is a homogeneous quadratic function of the velocities  Quadratic Contains linear Independent of terms terms generalized velocities If the generalized co- ordinate system in the qj’s is referred to as an orthogonal system. The special case where time does not appear explicitly in the transformation equations, and therefore bj = c = 0, and Eq. (3.42) reduces to That is, K.E. is a homogeneous quadratic function of the generalized velocities  GENERALIZED MOMENTUM Consider the motion of a particle of mass m moving along x-axis. Its linear momentum p is Kinetic energy Differentiating T with respect to For a system described by a set of generalized co-ordinates q1, q2,..., qn, we define generalized momentum pi corresponding to generalized co-ordinate qi as Sometimes it is also known as conjugate momentum (conjugate to coordinate qi ). In general, generalized momentum is a function of the q’s, ’s and t.. As the Lagrangian is utmost quadratic in the ’s, pi is a linear function of the ’s. The generalized momentum pi need not always have the dimension of linear momentum. However, the product of any generalized momentum and the associated co-ordinate must always have the dimension of angular momentum For a conservative system, the use of the expression for generalized momentum, Eq. (3.52), reduces Lagrange’s equations of motion to  Cyclic Co-ordinates Co-ordinates that do not appear explicitly in the Lagrangian of a system (although it may contain the corresponding generalized velocities) are said to be cyclic or ignorable. If qi is a cyclic co- ordinate In such a case and Lagrange’s equation reduces to which means that The generalized momentum conjugate to a cyclic co-ordinate is conserved during the motion.  Homogeneity of Time and Conservation of Energy Homogeneity in time implies that the Lagrangian of a closed system does not depend explicitly on the time t. That is, The total time derivative of the Lagrangian is That is, the quantity in parenthesis must be constant in time. Denoting the constant by H called the Hamiltonian of the system  It can be shown that H is the total energy of the system if (i) the potential energy V is velocity-independent and (ii) the transformation equations connecting the rectangular and generalized coordinates do not depend on time explicitly When condition (ii) is satisfied, the kinetic energy T is a homogeneous quadratic function of the generalized velocities and by Euler’s theorem, When condition (ii) is not satisfied, the Hamiltonian H is no longer equal to the total energy of the system. However, the total energy is still conserved for a conservative system. In the present case, T is a homogeneous quadratic function of the generalized velocities Problems A simple pendulum has a bob of mass m with a mass m1 at the moving support (pendulum with moving support) which moves on a horizontal line in the vertical plane in which the pendulum oscillates. Find the Lagrangian and Lagrange’s equation of motion. This pendulum has two degrees of freedom, and x and  can be taken as the generalized co-ordinates. Taking the point of support as the zero of potential energy Small angle, Sin =0 & Cos =1, M=m1 Show that the shortest distance between two points is a straight line. In a plane, element of arc length Find Lagrange’s equation of motion of the bob of a simple pendulum. l is a constant, kinetic energy of the bob Taking the mean position of the bob as the reference point L=T-V A rigid body capable of oscillating in a vertical plane about a fixed horizontal axis is called a compound pendulum. (i) Set up its Lagrangian; (ii) Obtain its equations of motion; and (iii) Find the period of the pendulum. Let the vertical plane of oscillation be xy. Let the point O be the axis of oscillation, m be the mass of the body, G its centre of mass and I its moment of inertia about the axis of oscillation. The system has only one degree of freedom. (i) Angle q can be taken as the generalized co- ordinate When the displacement is q, the kinetic energy With respect to the point of oscillation, the potential energy A mass M is suspended from a spring of mass m and spring constant k. Write the Lagrangian of the system and show that it executes S.H.M. in the vertical direction. Also, obtain an expression for its period of oscillation. The direction of motion of the mass is selected as the x-axis. The velocity of the spring at the end where the mass M is attached is maximum, say and minimum (zero) at x = 0. At the distance t from the fixed end, the velocity is where l is the length of the spring. If r is the mass per unit length of the spring, the kinetic energy of the element of length dt is For the whole spring Lagrange’s equation is which is the equation of S.H.M.