{"id":5,"date":"2018-11-02T04:57:59","date_gmt":"2018-11-02T04:57:59","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/2018\/11\/02\/chapter-1\/"},"modified":"2019-05-01T09:49:52","modified_gmt":"2019-05-01T09:49:52","slug":"constraints-and-generalised-coordinates","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/constraints-and-generalised-coordinates\/","title":{"rendered":"Constraints and Generalised Coordinates"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
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  1. Introduction:<\/strong><\/li>\r\n<\/ol>\r\n \r\n

    Classical mechanics deals with the description of a dynamical system which evolves in a three dimensional Euclidean space. The space and time are considered absolute and immutable entities. In the Newtonian world view, motion is described by Newton\u2019s laws of motion which are valid in an inertial frame of reference. The dynamical system may be a point particle, a rigid body or a collection of particles. Classical mechanics was developed by Newton. An alternative and attractive formulation was developed by Lagrange, Euler, Hamilton, Poisoon, Jacobi and others. This formulation made the transition from classical mechanics to quantum mechanics easier and found important role in the theory of classical and quantum fields.<\/p>\r\n\r\n

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    1. Constraints<\/strong><\/li>\r\n<\/ol>\r\n \r\n

      In real physical situations the motion is often constrained to move in a way such that some of its coordinates or velocity components satisfy certain relations all through its motion. The relations can be expressed in the form of equations or inequalities. For example, the motion of a particle on a circle or on an ellipse in the X-Y plane satisfies<\/p>\r\n \r\n\r\n\"\"\r\n

      if it is moving on a circle of radius a or on an ellipse of semi-major axis a and semi-minor axis b. The coordinates of a particle confine to move within a sphere of radius a satisfies<\/p>\r\n\"\"\r\n

      The constrained motion results on account of certain forces called \u2018constraining forces\u2019 which arise when the particle is in contact with the surface or the curve on which it is constrained to move. Constraints can be classified in different classes depending on their nature \u2018Holonomic\u2019<\/strong> constraints can be expressed in terms if algebraic equations in the form<\/p>\r\n \r\n\r\n\"\"\r\n

      and they can be made independent of velocities. If the constraints depend on velocities and cannot be expressed in \u2018integrable equations\u2019<\/strong> they are termed \u2018Non holonomic\u2019.<\/strong> Non-holonomic constraints may involve frictional forces or expressed in terms of an inequality or in terms of non-integrable equations like . The constraints that do not depend on time explicitly are called \u2018Scleronomic\u2019<\/strong> and Rheonomic<\/strong>\u2019 if they depend explicitly on time. Further the constraints are said to be \u2018conservative\u2019 if the total mechanical energy during the motion is conserved and \u2018dissipative\u2019 when the constraint forces do work and the mechanical energy is not conserved.<\/p>\r\n \r\n\r\nExamples:<\/strong>\r\n

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      1. Constraint equation satisfied by a particle moving on the surface of a sphere of radius a is<\/li>\r\n<\/ol>\r\n\"\"\r\n

        2. Constraints satisfied by the motion of an atom of radius a moving in a rectangular cavity of size are<\/p>\r\n \r\n\r\n\"\"\r\n

        In view of the fact that the coordinates of a constrained system satisfy certain relations, the number of independent variables required to fix the position and configuration of a dynamical system is reduced in<\/p>\r\n \r\n

        A free particle moving in a three dimensional Euclidean space has three degrees of freedom ( , , ) in Cartesian coordinates each varying between \u2212\u221e to + \u221e . N free particles likewise have 3N degrees of freedom.<\/p>\r\n \r\n

        A diatomic molecule having two identical atoms joined together by electromagnetic forces such that the bond length remains fixed has 3 x 2 \u2013 1=5 degrees of freedom. Each atom has 3 degrees of freedom with one constraint \u2212 =<\/p>\r\n \r\n

        A rigid body : A Rigid body is mathematically idealized as a system with large number of particles with fixed distances among themselves has 6 degrees of freedom. This is because any three points in a rigid body that are not collinear have 3 x 2 = 6 degrees of freedom and once any three arbitrary points in the rigid body are fixed, the rigid body is completely fixed.<\/span><\/p>\r\n \r\n

        How to choose the degrees of freedom?<\/p>\r\n \r\n

        There is a choice in choosing degrees of freedom in terms of<\/p>\r\n \r\n\r\ni) choice of the origin\r\n\r\n \r\n\r\nii) coordinate system (can be Cartesian, cylindrical, spherical etc.)\r\n\r\n \r\n

        Simple Example of constrained motion<\/strong><\/p>\r\n \r\n

        Example 1. A Block sliding without friction on an inclined plane<\/strong><\/p>\r\n \r\n\r\n\"\"\r\n

        Coordinate system: X-axis is pointing along the surface of the plane, Y-axis normal to the plane.The\u00a0 Block\u00a0 is confined to slide\u00a0 without\u00a0 friction along the\u00a0 plane,\u00a0 therefore\u00a0 the\u00a0 constraint\u00a0 is y = constant ne motion along Y-axis. The equations of motion are<\/p>\r\n\"\"\r\n

        Example 2. Atwoods\u2019s Machine.<\/strong><\/p>\r\n \r\n\r\nTwo\u00a0 masses\u00a0 connected\u00a0 by\u00a0 a\u00a0 mass\u00a0 less\u00a0 inextensible\u00a0 string\u00a0 of\u00a0 fixed\u00a0 length\u00a0 passing\u00a0 over\u00a0 a\u00a0<\/span>frictionless pulley.<\/span>Coordinate system: Since there is only vertical motion we use Y \u2013 coordinate in the down\u00a0<\/span>direction.<\/span>\r\n\r\n \r\n\r\nEquation\u00a0<\/span>of motion of\u00a0 1 and\u00a0 2 are given by<\/span>\r\n\r\n \r\n\r\n1\u00a0 1 = \u2212 1\u00a0 \u00a0+\u00a0 ;\u00a0 2\u00a0 2 = \u2212 2\u00a0 +\r\n\r\n \r\n\r\nWhere T is the tension in the string. Since the length of the string is fixed\r\n\r\n \r\n\r\n1 +\u00a0 \u00a02 = \u2212\u00a0 \u00a0. .\u00a0 2 = \u2212\u00a0 \u2212\u00a0 \u00a01\r\n\r\n \r\n\r\nSo\r\n\r\n \r\n\r\n\"\"\r\n\r\nGeneralised cordinates:<\/strong>\r\n\r\n \r\n

        We saw that for a system of N particles if we have k independent constraint equations, the number of degrees of freedom is 3N \u2013 k. The dynamical behavior of the system depends only on the coordinates corresponding to the degrees of freedom. Since these are fewer degrees of freedom then the position coordinates, required to specify the position of each particle in the system, can we eliminate the unnecessary coordinates? This will undoubtedly simplify the analysis of the motion particularly if we choose coordinates that take into account the constraints and are independent.<\/p>\r\n \r\n

        For holonomic constraints it is indeed possible to define a set of 3N-k independent coordinates called \u2018Generalized Coordinates\u2019 \u00a0that specify the motion of the system subject to the given constraints and which are independent of each other. The Cartesian coordinates can then be expressed in terms of known function of generalized coordinates \u00a0i.e.<\/p>\r\n \r\n\r\n\"\"\r\n

        In terms of components<\/p>\r\n \r\n\r\n\"\"\r\n

        where . It may or may not be always possible to have an analytical expression for these functions. The generalized coordinates eliminate the constraints forces from the problem and they are independent. The generalized coordinates need not be Cartesian and can be chosen as suitable to the problem considered. For example, the motion of a particle moving on an ellipse<\/p>\r\n\"\"\r\n

        has only one degree of freedom which can be chosen to be the polar angle \u00a0and the particle coordinates X and Y are expressible in terms of the generalized coordinates \u00a0as<\/p>\r\n\"\"\r\n

        for which the constraint equation is automatically satisfied.<\/p>\r\n \r\n

        Just as the velocity components \u00a0are defined for poition coordinates, we can define generalized velocity<\/p>\r\n \r\n\r\n\"\"\r\n

        Velocities being defined as total time derivative of the said coordinate. For a function \u00a0the total time derivative is given by the chain rule:<\/p>\r\n \r\n\r\n\"\"\r\n

        The use of generalized coordinates in place of position coordinates of the system was to eliminate the nondynamical degrees of freedom from the system. In a similar manner, constraint forces can be eliminated leaving only the generalized forces.<\/p>\r\n \r\n

        Let \u00a0be the force acting on the ith particle of a N particle system.\u00a0 The work done by the force for an arbitrary displacement \u00a0\u00a0is<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

        And the total work done on the system by all the forces is<\/p>\r\n \r\n\r\n\"\"\r\n

        Now RI \u2018s are expressible in terms of generalized coordinates as<\/p>\r\n\"\"\r\n\r\nThen\r\n\r\n\"\"\r\n\r\nSubstituting in (1.17)\r\n\r\n \r\n\r\n\"\"\r\n\r\nDefine\r\n\r\n\"\"\r\n

        as the Generalized force and the total work done on the system can now be written as<\/p>\r\n\"\"\r\n

        \"\"expressed only in terms of the generalized coordinates and forces.\u00a0 The expression denotes the work done by all generalized forces which equal to the number of degrees of freedom of the system in the arbitrary\u00a0displacement in generalized coordinates.\u00a0 Let us illustrate this with the example of Atwood\u2019s machine.<\/p>\r\n \r\n

        Choose the coordinates as shown \u00a0and \u00a0are the positions of masses \u00a0and \u00a0from a horizontal plane passing through the pulley. The y-axis points upwards since the length of the string is fixed and does not charge with time.<\/p>\r\n \r\n\r\n\"\"\r\n

        We can choose either \u00a0or \u00a0as generalized<\/p>\r\n \r\n

        Let us choose \u00a0as the generalized coordinates generalized velocity is given by<\/p>\r\n \r\n

        The Force \u00a0and acting on masses \u00a0and \u00a0are<\/p>\r\n \r\n\r\n\"\"\r\n\r\nNow\r\n\r\n \r\n\r\n\"\"\r\n

        Problem 1: How many degrees of freedom does the double pendulum has? Write down the constraint equations.<\/p>\r\n \r\n\r\n\"\"\r\n

        Problem 2. Four masses 1, 2, 3 and 4 are hanging such that they can move only in the vertical direction. How many degrees of freedom does it have? Write down the constraint equations.<\/p>\r\n\"\"\r\n\r\nSummary :<\/strong>\r\n