1) The document discusses canonical transformations in classical mechanics. Canonical transformations preserve the mathematical structure of Hamilton's equations of motion.
2) A canonical transformation involves changing the canonical coordinates (q,p) in phase space to new canonical coordinates (Q,P) while keeping the Hamilton's equations invariant.
3) Hamilton's principle is extended to phase space to derive the conditions for a transformation to be canonical. The action integral is written in terms of the Hamiltonian and the conditions of the transformation are that the variations of the action with respect to the new coordinates should vanish.
The document discusses the wave equation and its application to modeling vibrating strings and wind instruments. It describes how the wave equation can be separated into independent equations for time and position using the assumption that displacement is the product of separate time and position functions. This separation leads to trigonometric solutions that satisfy the boundary conditions of strings fixed at both ends. The solutions represent standing waves with discrete frequencies determined by the length, tension, and density of the string. Similar methods apply to wind instruments with different boundary conditions.
classification of second order partial differential equationjigar methaniya
This active learning assignment discusses the classification of second order partial differential equations. The general form of a non-homogeneous second order PDE is presented. A PDE is classified as elliptic if B^2 - 4AC < 0, parabolic if B^2 - 4AC = 0, and hyperbolic if B^2 - 4AC > 0. Three examples are worked through to demonstrate classifying PDEs as elliptic, parabolic, and hyperbolic by comparing them to the general form.
Applications of analytic functions and vector calculusPoojith Chowdhary
Analytic functions are functions that are locally defined by a convergent power series. They are used to compute cumulative, moving, centered, and reporting aggregates. Analytic functions are processed after joins, WHERE clauses, GROUP BY clauses, and HAVING clauses, and can only appear in the select list or ORDER BY clause. Common applications include calculating counts of employees under each manager.
This document provides an overview of complex analysis, including:
1) Limits and their uniqueness in complex analysis, such as the limit of a function f(z) as z approaches z0.
2) The definition of a continuous function in complex analysis as one where the limit exists at each point in the domain and equals the function value.
3) Analytic functions, which are differentiable in some neighborhood of each point in their domain.
This document discusses Hermite polynomials and their properties. It begins by introducing the Hermite equation, which arises from the theory of the linear harmonic oscillator. The Hermite equation is then solved using a series solution approach. Explicit formulas for the Hermite polynomials Hn(x) are derived for n=0,1,2,3,... using properties of the Hermite equation like recursion relations. Key properties of the Hermite polynomials like generating functions, even/odd behavior, Rodrigue's formula, orthogonality, and recurrence relations are also proved.
The document provides an overview of topics related to the Laplace transform and its applications. It defines the Laplace transform, discusses properties like linearity and examples of transforms of elementary functions. It also covers the inverse Laplace transform, differentiation and integration of transforms, evaluation of integrals using transforms, and applications to differential equations.
SERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATIONKavin Raval
This document discusses methods for solving ordinary differential equations (ODEs) using power series solutions and the Frobenius method. The power series method assumes solutions of the form of a power series centered at an ordinary point. The Frobenius method extends this to regular singular points by assuming solutions of the form of a power series multiplied by (x-x0)^r, where r is determined from the indicial equation. The document outlines the steps for both methods, which involve substituting the assumed series into the ODE and equating coefficients of like powers of (x-x0).
Series solutions at ordinary point and regular singular pointvaibhav tailor
The document discusses series solutions for second order linear differential equations near ordinary and regular singular points.
It defines an ordinary point as a point where the functions p(x) and q(x) in the normalized form of the differential equation are analytic. Near an ordinary point, there exist two linearly independent power series solutions of the form Σcn(x-a)n that converge within the radii of convergence of p(x) and q(x).
It also discusses finding series solutions near a regular singular point x0=0, where the limits of p(x) and q(x) as x approaches 0 exist. An initial guess of a power series solution with exponent r is made, and the
The document discusses the wave equation and its application to modeling vibrating strings and wind instruments. It describes how the wave equation can be separated into independent equations for time and position using the assumption that displacement is the product of separate time and position functions. This separation leads to trigonometric solutions that satisfy the boundary conditions of strings fixed at both ends. The solutions represent standing waves with discrete frequencies determined by the length, tension, and density of the string. Similar methods apply to wind instruments with different boundary conditions.
classification of second order partial differential equationjigar methaniya
This active learning assignment discusses the classification of second order partial differential equations. The general form of a non-homogeneous second order PDE is presented. A PDE is classified as elliptic if B^2 - 4AC < 0, parabolic if B^2 - 4AC = 0, and hyperbolic if B^2 - 4AC > 0. Three examples are worked through to demonstrate classifying PDEs as elliptic, parabolic, and hyperbolic by comparing them to the general form.
Applications of analytic functions and vector calculusPoojith Chowdhary
Analytic functions are functions that are locally defined by a convergent power series. They are used to compute cumulative, moving, centered, and reporting aggregates. Analytic functions are processed after joins, WHERE clauses, GROUP BY clauses, and HAVING clauses, and can only appear in the select list or ORDER BY clause. Common applications include calculating counts of employees under each manager.
This document provides an overview of complex analysis, including:
1) Limits and their uniqueness in complex analysis, such as the limit of a function f(z) as z approaches z0.
2) The definition of a continuous function in complex analysis as one where the limit exists at each point in the domain and equals the function value.
3) Analytic functions, which are differentiable in some neighborhood of each point in their domain.
This document discusses Hermite polynomials and their properties. It begins by introducing the Hermite equation, which arises from the theory of the linear harmonic oscillator. The Hermite equation is then solved using a series solution approach. Explicit formulas for the Hermite polynomials Hn(x) are derived for n=0,1,2,3,... using properties of the Hermite equation like recursion relations. Key properties of the Hermite polynomials like generating functions, even/odd behavior, Rodrigue's formula, orthogonality, and recurrence relations are also proved.
The document provides an overview of topics related to the Laplace transform and its applications. It defines the Laplace transform, discusses properties like linearity and examples of transforms of elementary functions. It also covers the inverse Laplace transform, differentiation and integration of transforms, evaluation of integrals using transforms, and applications to differential equations.
SERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATIONKavin Raval
This document discusses methods for solving ordinary differential equations (ODEs) using power series solutions and the Frobenius method. The power series method assumes solutions of the form of a power series centered at an ordinary point. The Frobenius method extends this to regular singular points by assuming solutions of the form of a power series multiplied by (x-x0)^r, where r is determined from the indicial equation. The document outlines the steps for both methods, which involve substituting the assumed series into the ODE and equating coefficients of like powers of (x-x0).
Series solutions at ordinary point and regular singular pointvaibhav tailor
The document discusses series solutions for second order linear differential equations near ordinary and regular singular points.
It defines an ordinary point as a point where the functions p(x) and q(x) in the normalized form of the differential equation are analytic. Near an ordinary point, there exist two linearly independent power series solutions of the form Σcn(x-a)n that converge within the radii of convergence of p(x) and q(x).
It also discusses finding series solutions near a regular singular point x0=0, where the limits of p(x) and q(x) as x approaches 0 exist. An initial guess of a power series solution with exponent r is made, and the
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
The document discusses line integrals in the complex plane. It defines line integrals, shows how complex line integrals are equivalent to two real line integrals, and reviews how to parameterize curves to evaluate line integrals. It also covers Cauchy's theorem, which states that the line integral of an analytic function around a closed curve in its domain is zero. The fundamental theorem of calculus for complex variables and the Cauchy integral formula are also summarized.
This document discusses Laplace transforms and their application to solving differential equations. It defines the Laplace transform, provides examples of common transform pairs, and lists several properties that allow transforms to be manipulated algebraically. The document states that Laplace transforms can convert differential equations into algebraic equations in the frequency domain, making them easier to solve. The transform method involves taking the Laplace transform of the differential equation, solving for the unknown variable, and taking the inverse Laplace transform to obtain the time domain solution.
The document discusses related rates problems. It begins by using resizing a rectangle on a computer screen as an example to demonstrate how the rates of change of the length (L) and width (W) relate to the rate of change of the area (A). The key steps are: (1) the area A is given by A=LW, (2) take the derivative of both sides, (3) use the product rule and chain rule to obtain A'=L'W+LW', (4) plug in the given rates of L' and W' to solve for A'.
The document then provides examples to demonstrate how to set up and solve related rates problems by translating the given rates into derivatives, applying
The document discusses Legendre functions, which are solutions to Legendre's differential equation. Legendre functions arise when solving Laplace's equation in spherical coordinates. Legendre polynomials were first introduced by Adrien-Marie Legendre in 1785 as coefficients in an expansion of Newtonian potential. The document covers topics such as Legendre polynomials, Rodrigues' formula, orthogonality of Legendre polynomials, and associated Legendre functions.
- The document discusses various types of power series representations of complex functions, including Taylor series, Maclaurin series, and Laurent series.
- It defines key concepts such as isolated singularities, classification of singularities into removable, pole, and essential types based on the principal part of the Laurent series, and the residue, which is the coefficient of the 1/(z-z0) term in the Laurent series at an isolated singularity z0.
- Examples are provided to illustrate these various types of series representations and singularities.
The one-dimensional wave equation governs vibrations of an elastic string. It is solved by separating variables, yielding solutions of the form F(x)G(t) where F and G satisfy ordinary differential equations. Boundary conditions require F(x) to be sinusoidal, with wavelengths that are integer multiples of the string length. The general solution is a superposition of these sinusoidal modes, with coefficients determined by the initial conditions. Motions of strings with different initial displacements are expressed as solutions to the one-dimensional wave equation.
First order non-linear partial differential equation & its applicationsJayanshu Gundaniya
There are five types of methods for solving first order non-linear partial differential equations:
I) Equations containing only p and q variables. II) Equations relating z as a function of u. III) Equations that can be separated into functions of single variables. IV) Clairaut's Form where the solution is directly substituted. V) Charpit's Method which is a general method taking integrals of auxiliary equations to solve dz=pdx+qdy and find the solution. These types cover a range of applications including Poisson's, Helmholtz's, and Schrödinger's equations in fields like electrostatics, elasticity, wave theory and quantum mechanics.
This document discusses dynamical systems. It defines a dynamical system as a system that changes over time according to fixed rules determining how its state changes from one time to the next. It then covers:
- The two parts of a dynamical system: state space and function determining next state.
- Classification of systems as deterministic/stochastic, discrete/continuous, linear/nonlinear, and autonomous/nonautonomous.
- Examples of discrete and continuous models, differential equations, and linear vs nonlinear systems.
- Terminology including phase space, phase curve, phase portrait, and attractors.
- Analysis methods including fixed points, stability, and perturbation analysis.
- Examples of harmonic oscillator,
This ppt covers the topic of B.Sc.1 Mathematics,unit - 5 , paper - 2, calculus- Introduction of Linear differential equation of second order , complete solution in terms of known integral belonging to the complementary function.
Central problem in mechanics is describing a system's mechanical state and how it evolves over time. Three formulations include Galileo/Newton using coordinates and velocities, Lagrange using generalized coordinates and velocities, and Hamilton using positions and momenta in phase space.
Time homogeneity leads to conservation of energy from the Lagrangian not explicitly depending on time. Space homogeneity leads to conservation of momentum from the Lagrangian being independent of position coordinates. Noether's theorem links symmetries like time and space homogeneity to conserved quantities like energy and momentum.
Controllability of Linear Dynamical SystemPurnima Pandit
The document discusses linear dynamical systems and controllability of linear systems. It defines dynamical systems as mathematical models describing the temporal evolution of a system. Linear dynamical systems are ones where the evaluation functions are linear. Controllability refers to the ability to steer a system from any initial state to any final state using input controls. The document provides the definition of controllability for linear time-variant systems using the controllability Gramian matrix. It also gives the formula for the minimum-norm control input that can steer the system between any two states. An example of checking controllability for a time-invariant linear system is presented.
1. The document discusses tensor analysis and its use in studying the Einstein field equations. It defines key tensors such as the Riemann-Christoffel curvature tensor and its properties including the antisymmetric and cyclic properties.
2. Bianchi identities are derived using a geodesic coordinate system. Taking the covariant derivative of the curvature tensor leads to the Bianchi identities.
3. Other concepts discussed include the Ricci tensor, gradient and divergence of tensors, and the Einstein tensor obtained by contracting the Bianchi identities. The Einstein tensor is related to the Ricci tensor and metric tensor.
The Fourier transform is a mathematical tool that transforms functions between the time and frequency domains. It breaks down any function or signal into the frequencies that make it up. This allows analysis of signals in the frequency domain, enabling applications like image and signal processing. The Fourier transform represents functions as a combination of sinusoidal functions like sines and cosines. The inverse Fourier transform reconstructs the original function from its frequency representation. Fourier transforms have many uses including solving differential equations, filtering sound and images, and analyzing signals like heartbeats.
The document discusses vector calculus concepts including:
1) Coordinate systems used in vector calculus problems including rectangular, cylindrical, and spherical coordinates.
2) How to write vectors and their components in each coordinate system.
3) Relationships between vectors in different coordinate systems using transformation matrices.
4) Concepts of gradient, divergence, and curl and their definitions and representations in different coordinate systems.
5) Theorems relating integrals, including the divergence theorem and Stokes' theorem.
1. Define the Gamma and Beta functions.
2. Express integrals involving products of powers of x with sine and cosine functions in terms of Beta functions.
3. State properties of Gamma and Beta functions including their relationship and formulas for computing their values.
This document discusses ideals and factor rings. It begins by defining an ideal as a subring of a ring where certain multiplication rules hold. It provides examples of ideals in various rings. It then discusses the existence of factor rings when a subring is an ideal. The elements of a factor ring are cosets of the ideal. It provides examples of computing operations in specific factor rings. Finally, it discusses prime and maximal ideals and their relationship to the structure of factor rings.
This presentation contributes towards understanding the periodic function of a Laplace Transform. A sum has been included to relate the method for this topic and a video also so that the learning can be easy.
1) The document describes a canonical transformation from the original (x, p) coordinates to new canonical coordinates (X, P) for the harmonic oscillator Hamiltonian.
2) An explicit transformation is found using a generating function of type 1. The new coordinates (X, P) correspond to an action-angle pair where P is the action (energy) and X is the cyclic coordinate proportional to time.
3) In the new coordinates, the Hamiltonian and equations of motion simplify greatly, with P being a constant of motion and X varying linearly with time.
The document provides an introduction to the Laplace transform and its applications in engineering analysis. It discusses key topics including:
- The history and development of the Laplace transform by mathematician Pierre-Simon Laplace.
- How the Laplace transform is used to transform a variable like time or position into a parameter to solve differential equations.
- Examples of using Laplace transform properties like linearity, shifting, and change of scale to calculate transforms of functions.
- How Laplace transforms can be used to solve ordinary and partial differential equations that model physical systems.
- Tables of common Laplace transforms to help with transforming functions.
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
The document discusses line integrals in the complex plane. It defines line integrals, shows how complex line integrals are equivalent to two real line integrals, and reviews how to parameterize curves to evaluate line integrals. It also covers Cauchy's theorem, which states that the line integral of an analytic function around a closed curve in its domain is zero. The fundamental theorem of calculus for complex variables and the Cauchy integral formula are also summarized.
This document discusses Laplace transforms and their application to solving differential equations. It defines the Laplace transform, provides examples of common transform pairs, and lists several properties that allow transforms to be manipulated algebraically. The document states that Laplace transforms can convert differential equations into algebraic equations in the frequency domain, making them easier to solve. The transform method involves taking the Laplace transform of the differential equation, solving for the unknown variable, and taking the inverse Laplace transform to obtain the time domain solution.
The document discusses related rates problems. It begins by using resizing a rectangle on a computer screen as an example to demonstrate how the rates of change of the length (L) and width (W) relate to the rate of change of the area (A). The key steps are: (1) the area A is given by A=LW, (2) take the derivative of both sides, (3) use the product rule and chain rule to obtain A'=L'W+LW', (4) plug in the given rates of L' and W' to solve for A'.
The document then provides examples to demonstrate how to set up and solve related rates problems by translating the given rates into derivatives, applying
The document discusses Legendre functions, which are solutions to Legendre's differential equation. Legendre functions arise when solving Laplace's equation in spherical coordinates. Legendre polynomials were first introduced by Adrien-Marie Legendre in 1785 as coefficients in an expansion of Newtonian potential. The document covers topics such as Legendre polynomials, Rodrigues' formula, orthogonality of Legendre polynomials, and associated Legendre functions.
- The document discusses various types of power series representations of complex functions, including Taylor series, Maclaurin series, and Laurent series.
- It defines key concepts such as isolated singularities, classification of singularities into removable, pole, and essential types based on the principal part of the Laurent series, and the residue, which is the coefficient of the 1/(z-z0) term in the Laurent series at an isolated singularity z0.
- Examples are provided to illustrate these various types of series representations and singularities.
The one-dimensional wave equation governs vibrations of an elastic string. It is solved by separating variables, yielding solutions of the form F(x)G(t) where F and G satisfy ordinary differential equations. Boundary conditions require F(x) to be sinusoidal, with wavelengths that are integer multiples of the string length. The general solution is a superposition of these sinusoidal modes, with coefficients determined by the initial conditions. Motions of strings with different initial displacements are expressed as solutions to the one-dimensional wave equation.
First order non-linear partial differential equation & its applicationsJayanshu Gundaniya
There are five types of methods for solving first order non-linear partial differential equations:
I) Equations containing only p and q variables. II) Equations relating z as a function of u. III) Equations that can be separated into functions of single variables. IV) Clairaut's Form where the solution is directly substituted. V) Charpit's Method which is a general method taking integrals of auxiliary equations to solve dz=pdx+qdy and find the solution. These types cover a range of applications including Poisson's, Helmholtz's, and Schrödinger's equations in fields like electrostatics, elasticity, wave theory and quantum mechanics.
This document discusses dynamical systems. It defines a dynamical system as a system that changes over time according to fixed rules determining how its state changes from one time to the next. It then covers:
- The two parts of a dynamical system: state space and function determining next state.
- Classification of systems as deterministic/stochastic, discrete/continuous, linear/nonlinear, and autonomous/nonautonomous.
- Examples of discrete and continuous models, differential equations, and linear vs nonlinear systems.
- Terminology including phase space, phase curve, phase portrait, and attractors.
- Analysis methods including fixed points, stability, and perturbation analysis.
- Examples of harmonic oscillator,
This ppt covers the topic of B.Sc.1 Mathematics,unit - 5 , paper - 2, calculus- Introduction of Linear differential equation of second order , complete solution in terms of known integral belonging to the complementary function.
Central problem in mechanics is describing a system's mechanical state and how it evolves over time. Three formulations include Galileo/Newton using coordinates and velocities, Lagrange using generalized coordinates and velocities, and Hamilton using positions and momenta in phase space.
Time homogeneity leads to conservation of energy from the Lagrangian not explicitly depending on time. Space homogeneity leads to conservation of momentum from the Lagrangian being independent of position coordinates. Noether's theorem links symmetries like time and space homogeneity to conserved quantities like energy and momentum.
Controllability of Linear Dynamical SystemPurnima Pandit
The document discusses linear dynamical systems and controllability of linear systems. It defines dynamical systems as mathematical models describing the temporal evolution of a system. Linear dynamical systems are ones where the evaluation functions are linear. Controllability refers to the ability to steer a system from any initial state to any final state using input controls. The document provides the definition of controllability for linear time-variant systems using the controllability Gramian matrix. It also gives the formula for the minimum-norm control input that can steer the system between any two states. An example of checking controllability for a time-invariant linear system is presented.
1. The document discusses tensor analysis and its use in studying the Einstein field equations. It defines key tensors such as the Riemann-Christoffel curvature tensor and its properties including the antisymmetric and cyclic properties.
2. Bianchi identities are derived using a geodesic coordinate system. Taking the covariant derivative of the curvature tensor leads to the Bianchi identities.
3. Other concepts discussed include the Ricci tensor, gradient and divergence of tensors, and the Einstein tensor obtained by contracting the Bianchi identities. The Einstein tensor is related to the Ricci tensor and metric tensor.
The Fourier transform is a mathematical tool that transforms functions between the time and frequency domains. It breaks down any function or signal into the frequencies that make it up. This allows analysis of signals in the frequency domain, enabling applications like image and signal processing. The Fourier transform represents functions as a combination of sinusoidal functions like sines and cosines. The inverse Fourier transform reconstructs the original function from its frequency representation. Fourier transforms have many uses including solving differential equations, filtering sound and images, and analyzing signals like heartbeats.
The document discusses vector calculus concepts including:
1) Coordinate systems used in vector calculus problems including rectangular, cylindrical, and spherical coordinates.
2) How to write vectors and their components in each coordinate system.
3) Relationships between vectors in different coordinate systems using transformation matrices.
4) Concepts of gradient, divergence, and curl and their definitions and representations in different coordinate systems.
5) Theorems relating integrals, including the divergence theorem and Stokes' theorem.
1. Define the Gamma and Beta functions.
2. Express integrals involving products of powers of x with sine and cosine functions in terms of Beta functions.
3. State properties of Gamma and Beta functions including their relationship and formulas for computing their values.
This document discusses ideals and factor rings. It begins by defining an ideal as a subring of a ring where certain multiplication rules hold. It provides examples of ideals in various rings. It then discusses the existence of factor rings when a subring is an ideal. The elements of a factor ring are cosets of the ideal. It provides examples of computing operations in specific factor rings. Finally, it discusses prime and maximal ideals and their relationship to the structure of factor rings.
This presentation contributes towards understanding the periodic function of a Laplace Transform. A sum has been included to relate the method for this topic and a video also so that the learning can be easy.
1) The document describes a canonical transformation from the original (x, p) coordinates to new canonical coordinates (X, P) for the harmonic oscillator Hamiltonian.
2) An explicit transformation is found using a generating function of type 1. The new coordinates (X, P) correspond to an action-angle pair where P is the action (energy) and X is the cyclic coordinate proportional to time.
3) In the new coordinates, the Hamiltonian and equations of motion simplify greatly, with P being a constant of motion and X varying linearly with time.
The document provides an introduction to the Laplace transform and its applications in engineering analysis. It discusses key topics including:
- The history and development of the Laplace transform by mathematician Pierre-Simon Laplace.
- How the Laplace transform is used to transform a variable like time or position into a parameter to solve differential equations.
- Examples of using Laplace transform properties like linearity, shifting, and change of scale to calculate transforms of functions.
- How Laplace transforms can be used to solve ordinary and partial differential equations that model physical systems.
- Tables of common Laplace transforms to help with transforming functions.
This document discusses using group theory and Lie algebras to formulate quantum mechanics from classical mechanics. It begins by reviewing classical phase space methods and their relation to Lie groups. It then develops an analogous formalism for quantum mechanics by replacing classical observables with operators satisfying the same Lie algebra. Unitary representations of this algebra define quantum states. The Heisenberg algebra is introduced for a particle, and its representation leads to a probabilistic interpretation. Dynamics are discussed using Hamiltonians of Newtonian form. As an example, the position-momentum uncertainty principle is derived from the Heisenberg commutation relation.
This document provides an overview of analog control systems and Laplace transforms. It introduces key concepts like Laplace transforms, common time domain inputs, transfer functions, and modeling electrical, mechanical and electromechanical systems using block diagrams and mathematical models. Examples are provided to illustrate Laplace transforms, transfer functions, and analyzing system response using poles, zeros and stability analysis.
This document provides an overview of analog control systems and Laplace transforms. It introduces key concepts like Laplace transforms, common time domain inputs, transfer functions, and modeling electrical, mechanical and electromechanical systems using block diagrams and mathematical models. Examples are provided to illustrate Laplace transforms, transfer functions, and analyzing system response using poles, zeros and stability analysis.
The document describes transfer function models for non-interacting and interacting systems. For a non-interacting system of two tanks in series, the transfer function relating the input flow to the first tank to the output flow of the second tank is a second-order model with two real poles. For an interacting system of two tanks where the liquid levels affect each other, the transfer function is also second-order but has one real pole and one negative zero, making it overdamped. The interacting system has a more complex dynamic model compared to the non-interacting case.
This document discusses the Legendre transformation, which is used to convert between Lagrangian and Hamiltonian formulations of mechanics and between different thermodynamic potentials. It provides examples of how the Legendre transformation converts between variables in classical mechanics and thermodynamics while preserving physical quantities like energy. The transformation allows describing a physical system using different but related variables that provide an equivalent description of the system's behavior.
The Quine-McCluskey method is used to minimize Boolean functions expressed in sum-of-products form. It works by generating prime implicants from the minterms and then eliminating variables to find an essential prime implicant. The method uses the Boolean rule A + A' = 1 repeatedly. It can handle incompletely specified functions using don't care conditions. The main steps are: 1) generate prime implicants, 2) construct a prime implicant table, 3) reduce the table by removing essential prime implicants and applying row and column dominance, 4) solve the reduced table.
The document discusses key concepts in Laplace transforms including:
1) The Laplace transform is defined as an integral transform that transforms a function of time into a function of a complex variable, simplifying analysis of differential equations.
2) Important properties include the Laplace transforms of derivatives and integrals, which allow transforming differential equations into algebraic equations.
3) The existence theorem guarantees a unique solution to initial value problems under certain conditions on the function.
Laplace Transformation & Its ApplicationChandra Kundu
This document presents an overview of the Laplace transform and its applications. It begins with an introduction to Laplace transforms as a mathematical tool to convert differential equations into algebraic expressions. It then provides definitions and properties of both the Laplace transform and its inverse. Examples are given of how Laplace transforms can be used to solve ordinary and partial differential equations, as well as applications in electrical circuits and other fields. The document concludes by noting some limitations of the Laplace transform method and references additional resources.
Statistical approach to quantum field theorySpringer
- The document discusses path integral formulations in quantum and statistical mechanics. It introduces Feynman's path integral approach, which sums over all possible quantum mechanical paths between two points, in contrast to matrix and wave mechanics formulations.
- It derives the Feynman-Kac formula, which provides a path integral representation for the quantum mechanical propagator through an application of Trotter's theorem. This sums over all broken-line paths between the initial and final points.
- It also discusses Kac's formula, which applies to positive operators and is used in the Euclidean formulation of quantum mechanics and statistical physics.
This document provides an overview of linear algebra, ordinary differential equations, and integral transforms taught in a course at National University of Sciences & Technology. It introduces the Laplace transform, a method for solving initial value problems by transforming differential equations into algebraic equations. Examples show how to take the Laplace transform of basic functions and use properties like shifting to solve problems. The document also discusses the inverse Laplace transform and applications of the method.
This document provides an overview of discrete-time systems and models. It begins by explaining that digital control involves systems that are updated at discrete time instants, which can be described by discrete-time models. The chapter objectives are then listed, which include explaining difference equations, obtaining z-transforms, solving difference equations, and more. An example is provided to illustrate how a discrete-time model arises from an analog system with piecewise constant control. Key concepts such as difference equations, the z-transform, and its properties are then defined and discussed.
PDE Constrained Optimization and the Lambert W-FunctionMichael Maroun
This document discusses using the Lambert W-function to solve partial differential equations (PDEs) that arise from optimization problems with PDE constraints. It first presents an example optimization problem involving minimizing the difference between a disk's actual and ideal temperature distributions. Solving the resulting PDE system may involve the Lambert W-function. It then briefly introduces the Lambert W-function and shows how it relates solutions of linear and nonlinear differential equations. Finally, it presents another nonlinear PDE whose solution involves the Lambert W-function.
The Laplace Transform of Modeling of a Spring-Mass-Damper System Mahmoud Farg
This document summarizes the use of Laplace transforms to model a spring-mass-damper system. It presents the equations of motion for a single mass connected to a spring and damper. Taking the Laplace transform of these second order differential equations allows them to be solved algebraically for various initial conditions. The document works through an example problem, applying the specific parameters of a mass, spring constant, and damping coefficient to determine the position of the mass over time. It concludes by discussing how to check the results and apply the final value theorem to find the steady state position.
This document provides information about the Laplace transform of periodic functions. It discusses the Laplace transform of periodic square waves, triangular waves, sawtooth waves, and staircase functions. It also discusses the Laplace transform of full-wave and half-wave rectification of a sine wave. Examples are provided to show how to calculate the Laplace transform of each of these periodic and rectified functions.
Decay Property for Solutions to Plate Type Equations with Variable CoefficientsEditor IJCATR
In this paper we consider the initial value problem for a plate type equation with variable coefficients and memory in
1 n R n ), which is of regularity-loss property. By using spectrally resolution, we study the pointwise estimates in the spectral
space of the fundamental solution to the corresponding linear problem. Appealing to this pointwise estimates, we obtain the global
existence and the decay estimates of solutions to the semilinear problem by employing the fixed point theorem
1) The document discusses concepts related to differential equations, including defining a differential equation as an equation that relates the time derivative of a variable to its level.
2) It provides an example differential equation and classifies its variables as endogenous, exogenous, and predetermined.
3) The document analyzes the steady state of the example differential equation, finding the steady state value by setting the time derivative equal to zero and discussing its stability properties.
This document discusses the equivalence of non-deterministic finite automata (NFAs) and deterministic finite automata (FAs), and proves that:
1) The languages accepted by NFAs are regular languages.
2) NFAs and FAs have the same computation power and accept the same regular languages.
3) Any NFA can be converted to an equivalent FA, by constructing a FA where the states are power sets of the NFA states and the transitions mimic the NFA transitions.
chapter-2.ppt control system slide for studentslipsa91
This document discusses mathematical models of physical systems and control systems. It introduces differential equations that describe the behavior of mechanical, hydraulic, and electrical systems. Since most physical systems are nonlinear, the document discusses linearization approximations that allow the use of Laplace transform methods to analyze input-output relationships and design control systems. Block diagrams are presented as a convenient tool for analyzing complicated control systems.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
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Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
2. Canonical Variables and Hamiltonian Formalism
are independent variables in phase space on equal footing
As we have seen, in the Hamiltonian Formulation of Mechanics,
j j
j j
H H
q and p
p q
- As long as the new variables formally satisfy this abstract structure (the
form of the Hamilton’s Equations.
,
j j
q p
The Hamilton’s Equation for are “symmetric” (symplectic, later)
,
j j
q p
This elegant formal structure of mechanics affords us the freedom in
selecting other (may be better) canonical variables as our phase space
“coordinates” and “momenta”
2
3. Canonical Transformation
Recall (from hw) that the Euler-Lagrange Equation is invariant for a point
transformation:
Now, the idea is to find a generalized (canonical) transformation in phase
space (not config. space) such that the Hamilton’s Equations are invariant !
( , )
j j
Q Q q t
i.e., if we have, 0,
j j
L d L
q dt q
then, 0,
j j
L d L
Q dt Q
( , , )
( , , )
j j
j j
Q Q q p t
P P q p t
(In general, we look for
transformations which
are invertible.)
3
4. Invariance of EL equation for Point Transformation
Given:
From the inverse point transformation equation , we have,
( , )
j j
Q Q q t
and a point transformation:
0,
j j
d L L
dt q q
Formally, calculate: i i
i i
j i j i j
q q
L L L
Q q Q q Q
i i
j
i
i i
i
j j
q
Q
q
L L
Q
L
Q q q
(chain rule)
( , )
i i
q q Q t
0
i
j
q
Q
and i
j
i
j
q
Q
Q
q
i i
i k
k k
q q
q Q
Q t
First look at the situation in config. space first:
Need to show: 0
j j
d L L
dt Q Q
4
5. Invariance of EL equation for Point Transformation
Forming the LHS of EL equation with :
j j
d L L
LHS
dt Q Q
i i i
i i i
i j i j i j
d
dt
q q q
L L L
LHS
q Q q Q q Q
j
Q
i i
i i i
i j i j
i i
i j i j
q q
L d d L
q dt Q dt
q q
L L
q Q q Q
q Q
i i
i i i i
i
i
j
j j
i
j
L L
q
q q
d L L d
dt q q dt Q
q q
Q Q q Q
5
6. Invariance of EL equation for Point Transformation
i
i i
i i
i
i j j
j
q q
d L L d
LHS
dt q q
L
Q
q d Q
q
t
Q
0
(Since that’s what given !)
0
i i
j j
i i
j j
dq q
Q dt Q
q q
Q Q
(exchange order of diff)
0
LHS 0
j j
d L L
dt Q Q
6
7. Canonical Transformation
Now, back to phase space with q’s & p’s, we need to find the appropriate
(canonical) transformation
with which the Hamilton’s Equations are satisfied:
such that there exist a transformed Hamiltonian
( , , ) and ( , , )
j j j j
Q Q q p t P P q p t
( , , )
K Q P t
j j
j j
K K
Q and P
P Q
(The form of the EOM must be invariant in the new coordinates.)
** It is important to further stated that the transformation considered
must also be problem-independent meaning that must be canonical
coordinates for all system with the same number of dofs.
( , )
Q P
7
8. Canonical Transformation
To see what this condition might say about our canonical transformation,
we need to go back to the Hamilton’s Principle:
Hamilton’s Principle: The motion of the system in configuration space is
such that the action I has a stationary value for the actual path, .i.e.,
2
1
0
t
t
I Ldt
Now, we need to extend this to the 2n-dimensional phase space
1. The integrant in the action integral must now be a function of the
independent conjugate variable and their derivatives
2. We will consider variations in all 2n phase space coordinates
,
j j
q p ,
j j
q p
8
9. Hamilton’s Principle in Phase Space
1. To rewrite the integrant in terms of , we will utilize the
definition for the Hamiltonian (or the inverse Legendre Transform):
2 2
1 1
( , , ) 0
t t
j j
t t
I Ldt p q H q p t dt
Substituting this into our variation equation, we have
2. The variations are now for n and n : (all q’s and p’s are independent)
'
j
q s
, , ,
j j j j
q p q p
( , , )
j j j j
H p q L L p q H q p t
(Einstein’s sum rule)
'
j
p s
again, we will required the variations for the to be zero at ends
j
q
The rewritten integrant is formally a
function of but in fact it does not depend on , i.e.
This fact will proved to be useful later on.
( , , ) ( , , )
j j
q q p p q H q p t
, , ,
j j j j
q p q p
j
p
0
j
p
9
10. Hamilton’s Principle in Phase Space
Affecting the variations on all 2n variables , we have,
,
j j
q p
'
j
p s
2
1
0
t
j j
j j j
t
j j
j j j
q q
I
d d d
q q
p p
d d dt
p p
'
j
q s
As in previous discussion, the second term in the sum for can be
rewritten using integration by parts:
'
j
q s
2
2
2
1
1 1
t
t
t
j j j
j j j
t
t t
q q q d
dt dt
q q dt q
0
0
j j j
j j j
q q
p p
10
11. Hamilton’s Principle in Phase Space
Previously, we have required the variations for the to be zero at end pts
j
q
So, the first sum with can be written as:
'
j
q s
1 2
,
0
j
t t t
q
so that,
2
1
where
t
j
j j
j j j
t
q
d
q dt q d
q dt q
2
1
t
j j
j j j
t
q q d
d d dt
q dt q
2
2
1
1
t
t
j j
j j t
t
q q
dt
q q
2
1
t
j
j
t
q d
dt
dt q
11
12. Note, since , without enforcing the
Hamilton’s Principle in Phase Space
'
j
p s
Now, perform the same integration by parts to the corresponding term for
j
p
0
j
p
2
1
0
t
j
j t
p
p
we have,
2
1
where
t
j
j j
j j j
t
p
d
p dt p d
p dt p
2
2
2
1
1 1
t
t
t
j j j
j j j
t
t t
p p p d
dt dt
p p dt p
variations for to be zero at end points.
This gives the result for the 2nd sum in the variation equation for :
'
j
p s
12
13. Hamilton’s Principle in Phase Space
Putting both terms back together, we have:
2
1
0
t
j j
j j
j j j j
t
I d d
d q p dt
q dt q p dt p
1 2
Since both variations are independent, and must vanish independently !
2
1
1 0
j j
d
dt q q
0
j
j
H
p
q
( , , ) ( , , )
j j
q q p p q H q p t
and
j
j j j
H
p
q q q
j
j
H
p
q
(one of the Hamilton’s
equations)
13
14. Hamilton’s Principle in Phase Space
2
1
0
t
j j
j j
j j j j
t
I d d
d q p dt
q dt q p dt p
1 2
2 0
j j
d
dt p p
0 0
j
j
H
q
p
( , , ) ( , , )
j j
q q p p q H q p t
0 and j
j j j
H
q
p p p
j
j
H
q
p
(2nd Hamilton’s
equations)
14
15. Hamilton’s Principle in Phase Space
So, we have just shown that applying the Hamilton’s Principle in Phase
Space, the resulting dynamical equation is the Hamilton’s Equations.
15
j
j
H
p
q
j
j
H
q
p
16. Hamilton’s Principle in Phase Space
Notice that a full time derivative of an arbitrary function F of can
be put into the integrand of the action integral without affecting the
variations:
Thus, when variation is taken, this constant term will not contribute !
( , , )
q p t
2
1
( , , )
t
j j
t
dF
p q H q p t dt
dt
2
2
1
1
( , , )
t
t
j j
t
t
dF
p q H q p t dt dt
dt
2
2
1
1
t
t
t
t
const
dF F
16
17. Canonical Transformation
Now , we come back to the question: When is a transformation to
canonical?
,
Q P
( , , ) 0
j j
p q H q p t dt
This means that we need to have the following variational conditions:
We need Hamilton’s Equations to hold in both systems
( , , ) 0
j j
P Q K Q P t dt
AND
For this to be true simultaneously, the integrands must equal
And, from our previous slide, this is also true if they are differed by
a full time derivative of a function of any of the phase space variables
involved + time:
( , , ) ( , , ) , , , ,
j j j j
dF
p q H q p t P Q K Q P t q p Q P t
dt
17
18. Canonical Transformation
F is called the Generating Function for the canonical
transformation:
As the name implies, different choice of F give us the ability to
generate different Canonical Transformation to get to different
( , , ) ( , , ) , , , , (*) ( 9.11)
j j j j
dF
p q H q p t P Q K Q P t q p Q P t G
dt
( , , )
( , , )
j j
j j
Q Q q p t
P P q p t
18
( , ) , :
j j j j
q p Q P
,
j j
Q P
F is useful in specifying the exact form of the transformation if it
contains half of the old variables and half of the new variables. It,
then, acts as a bridge between the two sets of canonical variables.
19. Canonical Transformation
Depending on the form of the generating functions (which pair of
canonical variables being considered as the independent variables for
the Generating Function), we can classify canonical transformations
into four basic types.
F is called the Generating Function for the canonical
transformation:
( , , ) ( , , ) , , , , ( 9.11)
(*)
j j j j
dF
p q H q p t P Q K Q P t q p Q P t G
dt
( , , )
( , , )
j j
j j
Q Q q p t
P P q p t
19
( , ) , :
j j j j
q p Q P
20. Canonical Transformation: 4 Types
1
, ,
j
j
F
p q Q t
q
1
, ,
j
j
F
P q Q t
Q
1
F
K H
t
( , , ) ( , , ) , ,
j j j j
dF
p q H q p t P Q K Q P t old new t
dt
Type 1:
2
, ,
j
j
F
p q P t
q
2
, ,
j
j
F
Q q P t
P
2
F
K H
t
1( , , )
q
F Q
F t
Type 2:
2 ( , , ) i i
F F t
q Q P
P
3
, ,
j
j
F
q p Q t
p
3
, ,
j
j
F
P p Q t
Q
3
F
K H
t
Type 3:
3 ( , , ) i i
F F t
p q p
Q
4
, ,
j
j
F
q p P t
p
4
, ,
j
j
F
Q p P t
P
4
F
K H
t
Type 4:
4 ( , , ) i i i i
p
F F t q p P
P Q
20
21. Canonical Transformation: Type 1
Or, we can write the equation in differential form:
Type 1: | F is a function of q and Q + time
Writing out the full time derivative for F, Eq (1) becomes:
1
1
1
j j j
j j
j
j
j
F
H K
t
F
p q q
q
F
P Q Q
Q
1( , , )
F F q Q t
(again E’s
sum rule)
1 1 1
0
j
j
j
j
j j
F
dq dQ K H
F
p
q
t
P
Q
d
t
F
(old) (new)
21
(write out and multiply the equation by )
, and
j j
j j
dq dQ
q Q
dt dt
dt
22. Canonical Transformation: Type 1
Since all the are independent, their coefficients must vanish
independently. This gives the following set of equations:
For a given specific expression for , e.g.
1
, , ( 1)
j
j
F
p q Q t C
q
,
j j
q Q
1
, , ( 2)
j
j
F
P q Q t C
Q
1
( 3)
F
K H C
t
1 , ,
F q Q t
1 , , j j
F q Q t q Q
. ( 1)
Eq C
are n relations defining in terms of and they can
be inverted to get the 1st set of the canonical transformation:
j
p , ,
j j
q Q t
1
, ,
j j j j
j
F
p q Q t Q Q p
q
In the specific example, we have:
22
These are the equations in the Table 9.1 in the book.
23. Canonical Transformation: Type 1
1
, ,
j j j j
j
F
P q Q t q P q
Q
1
F
K H
t
K H
.( 2)
Eq C
are n relations defining in terms of . Together with
our results for the , the 2nd set of the canonical transformation
can be obtained.
j
P , ,
j j
q Q t
Again, in the specific example, we have:
j
Q
.( 3)
Eq C
gives the connection between K and H:
(note: is a function of the new variables so that the RHS needs
to be re-express in terms of using the canonical transformation.)
( , , )
K Q P t
,
j j
Q P
23
24. Canonical Transformation: Type 1
j j
P q
K H
In summary, for the specific example of a Type 1 generating function:
We have the following:
Note: this example results in basically swapping the generalized coordinates with
their conjugate momenta in their dynamical role and this exercise demonstrates
that swapping them basically results in the same situation !
j j
Q p
1 , , j j
F q Q t q Q
Canonical Transformation
and Transformed Hamiltonian
Emphasizing the equal role for q and p in Hamiltonian Formalism !
24
25. Canonical Transformation: Type 2
Substituting into our defining equation for canonical transformation, Eq. (1):
(One can think of as the Legendre transform of in
exchanging the variables Q and P.)
Type 2: , where F2 is a function of q and P + time
2 ( , , ) j j
F F q t Q P
P
Again, writing the equation in differential form:
2 2 2
0
j j j j
j j
F F F
p dq Q dP K H dt
q P t
(old) (new)
2
F ( , , )
F q Q t
25
j j j j
dF
p q H P Q K
dt
j j j j
p q H P Q
2 2 2
j j j j
j j
F F F
K q P PQ
t q P
j j
PQ
26. Canonical Transformation: Type 2
Since all the are independent, their coefficients must vanish
independently. This gives the following set of equations:
For a given specific expression for , e.g.
2
, ,
j
j
F
p q P t
q
,
j j
q P
2
, ,
j
j
F
Q q P t
P
2
F
K H
t
2 , ,
F q P t
2 , , j j
F q P t q P
2
, ,
j j
j
j j
F
p q P t P
q
P p
2
, ,
j j
j
j j
F
Q q P t q
P
Q q
K H
Thus, the identity transformation is also a canonical transformation !
26
27. Canonical Transformation: Type 2
Let consider a slightly more general example for type 2:
Going through the same procedure, we will get:
2 , , j j
F F q P t Q P
2
j
j
j j
j j
F
p
q
f g
p P
q q
2
1, ,
j
j
j n
F
Q
P
Q f q q t
j
f g
K H P
t t
Notice that the Q equation is the general point transformation in the
configuration space. In order for this to be canonical, the P and H
transformations must be handled carefully (not necessary simple functions).
with
2 1 1
, , , , , , , ,
n j n
F q P t f q q t P g q q t
where f and g are function of q’s only + time
27
28. Canonical Transformation: Summary
(Results are summarized in Table 9.1 on p. 373 in Goldstein .)
The remaining two basic types are Legendre transformation of the remaining
two variables:
3 ( , , ) j j
F F Q t p p
p q q
Canonical Transformations form a group with the following properties:
4 ( , , ) &
j j j j
F F t p q Q P q p
p P Q P
1. The identity transformation is canonical (type 2 example)
2. If a transformation is canonical, so is its inverse
3. Two successive canonical transformations (“product”) is canonical
4. The product operation is associative
28
29. Canonical Transformation: 4 Types
1
, ,
j
j
F
p t
q
q Q
1
, ,
j
j
F
P t
Q
q Q
1
F
K H
t
( , , ) ( , , ) , ,
j j j j
dF
p q H q p t P Q K Q P t old new t
dt
( , , )
( , , )
j j
j j
Q Q q p t
P P q p t
Type 1:
2
, ,
j
j
F
p t
q
q P
2
, ,
j
j
F
Q t
P
q P
2
F
K H
t
1( , , )
q
F Q
F t
Type 2:
2 ( , , ) i i
F F t
q Q P
P
3
, ,
j
j
F
q t
p
p Q
3
, ,
j
j
F
P t
Q
p Q
3
F
K H
t
Type 3:
3 ( , , ) i i
F F t
p q p
Q
4
, ,
j
j
F
q t
p
p P
4
, ,
j
j
F
Q t
P
p P
4
F
K H
t
Type 4:
4 ( , , ) i i i i
p
F F t q p P
P Q
var
ind
var
dep
29
30. Canonical Transformation (more)
If we are given a canonical transformation
How do we find the appropriate generating function F ?
- Let say, we wish to find a generating function of the 1st type, i.e.,
( , , )
(*)
( , , )
j j
j j
Q Q q p t
P P q p t
1( , , )
F F q Q t
(Note: generating function of the other types can be obtain through an
appropriate Legendre transformation.)
- Since our chosen generating function (1st type) depends on q, Q, and t
explicitly, we will rewrite our p and P in terms of q and Q using Eq. (*):
( , , )
j j
p p q Q t
( , , )
j j
P P q Q t
30
31.
1 1
, , , ,
j i
i i j j j j
P F q Q t F q Q t p
q q Q Q q Q
Canonical Transformation (more)
Now, from the pair of equations for the Generating Function Derivatives
(Table 9.1), we form the following diff eqs,
can then be obtained by directly integrating the above equations
and combining the resulting expressions.
1 , ,
( , , )
j j
j
F q Q t
p p q Q t
q
1( , , )
F q Q t
1 , ,
( , , )
j j
j
F q Q t
P P q Q t
Q
Note: Taking the respective partials of q and Q of the above equations,
31
32. Canonical Transformation (more)
Now, from the pair of equations for the Generating Function Derivatives
(Table 9.1), we form the following diff eqs,
can then be obtained by directly integrating the above equations
and combining the resulting expressions.
1 , ,
( , , )
j j
j
F q Q t
p p q Q t
q
1( , , )
F q Q t
1 , ,
( , , )
j j
j
F q Q t
P P q Q t
Q
Note: Since dF1 is an exact differential wrt q and Q, so the two exps are equal,
2 2
1 1
j
j i
i j
i j i
F F
q Q Q
P p
q
q Q
(We will give the full
list of relations later.)
32
33. Canonical Transformation (more)
Example (G8.2): We are given the following canonical transformation for a
system with 1 dof:
1( , )
F q Q
( , ) cos sin
( , ) sin cos
Q Q q p q p
P P q p q p
(Q and P is being rotated in phase space from q and p by an angle )
We seek a generating function of the 1st kind:
(HW: showing this
trans. is canonical)
33
First, notice that the cross-second derivatives for F1 are equal as required for a
canonical transformation:
1
s
cot
n in
si
1
F Q
p q
Q q Q Q
1
s
cot
n in
si
1
F q
P Q
q Q q q
34. Canonical Transformation (more)
Now, integrating the two partial differential equations:
2
1 cot
sin 2
Qq q
F h Q
Comparing these two expression, one possible solution for is,
2
1 cot
sin 2
qQ Q
F g q
2 2
1
1
, cot
sin 2
Qq
F q Q q Q
34
1
F
Rewrite the transformation in terms of q and Q (indep. vars of F1 ):
cos
sin cos
sin sin
Q
q q
cot
sin
Q
q
cot
sin
q
Q
1
( , )
F
p p q Q
q
1
( , )
F
P P q Q
Q
35. Canonical Transformation (more)
As we have discussed previously, we can directly use the fact that F2 is the
Legendre transform of F1,
1 2 ( , , ) j j
F F q P t Q P
Now, from the CT, we can write Q by q and P (F2 should be in q & P):
35
2 1
( , , ) ( , , )
F q P t F q Q t QP
2 2
2
1
, cot
sin 2
Qq
F q P q Q QP
cos sin
sin cos
Q q p
P q p
tan
cos
q
Q P
Now, let say we want to fine a Type-2 Generating function for this
problem…
2 ( , , )
P
F q t
36. Canonical Transformation (more)
As we have discussed previously, we can directly use the fact that F2 is the
Legendre transform of F1,
1 2 ( , , ) j j
F F q P t Q P
This then gives:
36
2 1
( , , ) ( , , )
F q P t F q Q t QP
2 2
2
1
, cot
sin 2
Qq
F q P q Q QP
2
2
2
1
, tan tan cot
sin cos 2 cos
q q q
F q P P P q P
Q
37. Canonical Transformation (more)
As we have discussed previously, we can directly use the fact that F2 is the
Legendre transform of F1,
1 2 ( , , ) j j
F F q P t Q P
37
2
2
2
1
, tan tan cot
sin cos 2 cos
q q q
F q P P P q P
2
2
2
tan
cos sin cos
qP q
P
2
2 2
1 2
cot tan
2 cos sin cos
q qP
q P
38. Canonical Transformation (more)
As we have discussed previously, we can directly use the fact that F2 is the
Legendre transform of F1,
1 2 ( , , ) j j
F F q P t Q P
38
2
2
2
1
, tan tan cot
sin cos 2 cos
q q q
F q P P P q P
2 2
cos sin cos
qP q
2
tan
P
2
2
1
cot
2 cos sin
q
q
2
cos
qP
2
tan
P
39. Canonical Transformation (more)
As we have discussed previously, we can directly use the fact that F2 is the
Legendre transform of F1,
1 2 ( , , ) j j
F F q P t Q P
39
2
2
2
1
, tan tan cot
sin cos 2 cos
q q q
F q P P P q P
2 2
2
1
, tan
cos 2
qP
F q P q P
Finally,
40. Canonical Transformation (more)
Alternatively, we can substitute F into Eq. (*)G9.11 , results in replacing the
term by in our condition for a canonical transformation,
2 ( , , ) j j
F F q P t Q P
j j
P Q
j j
Q P
j j j j
dF
p q H Q P K
dt
Recall, this procedure gives us the two partial derivatives relations for F2:
2 , ,
j
j
F q P t
p
q
2 , ,
j
j
F q P t
Q
P
40
[Or, use the Table]
41. Canonical Transformation (more)
2 ( , , )
F q P t
To solve for in our example, again, we rewrite our given canonical
transformation in q and P explicitly.
Integrating and combining give,
( , ) cos sin
( , ) sin cos
Q Q q p q p
P P q p q p
tan
cos
P
q
2
( , )
F
p p q P
q
sin
cos sin
cos cos
P
q q
tan
cos
q
P
2
( , )
F
Q Q q P
P
2 2
2
1
, tan
cos 2
qP
F q P q P
41
42. Notice that when ,
Canonical Transformation (more)
0
so that our coordinate transformation is just the identity
transformation: and
sin 0
Q q
P p
p, P CANNOT be written explicitly in terms of q and Q !
so our assumption for using the type 1 generating function
(with q and Q as indp var) cannot be fulfilled.
Consequently, blow up and cannot be used to derive the
canonical transformation:
2 2
1
1
, cot as 0
sin 2
Qq
F q Q q Q
1 ,
F q Q
( , ) cos sin
( , ) sin cos
Q Q q p q p
P P q p q p
But, using a Type 2 generating function will work.
42
43. Similarly, we can see that when ,
Canonical Transformation (more)
2
our coordinate transformation is a coordinate switch ,
2 2
2
1
, tan as 0
cos 2
qP
F q P q P
cos 0
Q p
P q
p, Q CANNOT be written explicitly in terms of q and P !
so the assumption for using the type 2 generating function
(with q and P as indp var) cannot be fulfilled.
Consequently, blow up and cannot be used to derive the
canonical transformation:
2 ,
F q P
( , ) cos sin
( , ) sin cos
Q Q q p q p
P P q p q p
But, using a Type 1 generating function will work in this case.
43
44. Canonical Transformation (more)
- A suitable generating function doesn’t have to conform to only one of
the four types for all the degrees of freedom in a given problem !
For a generating function to be useful, it should depends on half of
the old and half of the new variables
As we have done in the previous example, the procedure in solving
for F involves integrating the partial derivative relations resulted from
“consistence” considerations using the main condition for a canonical
transformation, i.e.,
- There can also be more than one solution for a given CT
( , , ) ( , , ) ( 9.11)
j j j j
dF
p q H q p t P Q K Q P t G
dt
- First , we need to choose a suitable set of independent variables for the
generating function.
44
45. Canonical Transformation (more)
For these partial derivative relations to be solvable, one must be
able to feed-in 2n independent coordinate relations (from the given
CT) in terms of a chosen set of ½ new + ½ old variables.
- In general, one can use ANY one of the four types of generating
functions for the canonical transformation as long as the RHS of the
transformation can be written in terms of the associated pairs of phase
space coordinates: (q, Q, t), (q, P, t), (q, Q, t), or (p, P, t).
- On the other hand, if the transformation is such that the RHS cannot
written in term of a particular pair: (q, Q, t), (q, P, t), (q, Q, t), or (p, P, t),
then that associated type of generating functions cannot be used.
1
T 2
T 3
T 4
T
45
46. Canonical Transformation: an example with two
dofs
- As we will see, this will involve a mixture of two different basic types.
2 2
2 2
(2 )
(2 )
Q p a
P q b
- To see in practice how this might work… Let say, we have the following
transformation involving 2 dofs:
1 1
1 1
(1 )
(1 )
Q q a
P p b
1 1 2 2 1 1 2 2
, , , , , ,
q p q p Q P Q P
46
47. - As an alternative, we can try to use the set as our independent
variables. This will give an F which is a mixture of Type 3 and 1.
Canonical Transformation: an example with two
dofs
- First, let see if we can use the simplest type (type 1) for both dofs, i.e., F
will depend only on the q-Q’s:
1 2 1 2
( , , , , )
F q q Q Q t
Notice that Eq (1a) is a relation linking only , they CANNOT
both be independent variables Type 1 (only) WON’T work !
1
1 2 2
, , ,
Q
p q Q
1 1
,
q Q
(In Goldstein (p. 377), another alternative was using resulted in a
different generating function which is a mixture of Type 2 and 1.)
1 2 1 2
, , ,
q q P Q
47
2 2
2 2
(2 )
(2 )
Q p a
P q b
1 1
1 1
(1 )
(1 )
Q q a
P p b
48. Now, with this set of ½ new + ½ old independent variable chosen, we need to
derive the set of partial derivative conditions by substituting
Canonical Transformation: an example with two
dofs
From our CT, we can write down the following relations:
Dependent variables
2
2
P q
1
1
q Q
1 2 1 2
( , , , , )
F p q Q Q t
1 2 1 2
, , ,
p q Q Q
Independent variables
1 2 1 2
, , ,
q p P P
2
2
p Q
1
1
P p
into Eq. 9.11 (or look them up from the Table).
(*)
48
49. we will use the following Legendre transformation:
Canonical Transformation: an example with two
dofs
The explicit independent variables (those appear in the differentials) in Eq.
9.11 are the q-Q’s. To do the conversion:
1 2 1 2 1 1
'( , , , , )
F F p q Q Q t q p
2 1 2
1
1 2 1 2
, , ,
, , ,
q
q
p
q Q Q
Q Q
(Eq. 9.11’s explicit ind vars)
(our preferred ind vars)
Substituting this into Eq. 9.11, we have:
1 1
p q
2 2 1 1 2 2
p q H PQ PQ
dF
dt
K
1 2
1 1 2 2 1 2 1 1 1 1
1 2 1 2
' ' ' '
P
F F F F
p q Q Q q p p
Q PQ K q
p q Q Q
'
F
t
49
50. 2 2 1 1 2 2 1 2 1 2 1 1
1 2 1 2
' ' ' ' '
F F F F F
p q H PQ PQ K p q Q Q q p
p q Q Q t
Canonical Transformation: an example with two
dofs
Comparing terms, we have the following conditions:
1
1
'
F
q
p
2
2
'
F
p
q
'
F
K H
t
1
1
'
F
P
Q
2
2
'
F
P
Q
As advertised, this is a mixture of Type 3 and 1 of the basic CT.
50
1 1 1
1
1
2 2 2 2
2
1
2
1
2 1
2
'
' '
'
'
F
p q q
q
F
PQ Q
F
PQ Q
Q
F
p
F
q
Q
p
H K
p t
51. Canonical Transformation: an example with two
dofs
Substituting our coordinates transformation [Eq. (*)] into the partial
derivative relations, we have :
1 1
1
'
F
q Q
p
2 2
2
'
F
p Q
q
1 1
1
'
F
P p
Q
2 2
2
'
F
P q
Q
1 1 2 1 2
' ( , , )
F Q p f q Q Q
2 2 1 1 2
' ( , , )
F Q q k p Q Q
1 1 1 2 2
' ( , , )
F p Q g p q Q
2 2 1 2 1
' ( , , )
F q Q h p q Q
1 1 2 2
'
F p Q q Q
(Note: Choosing instead, Goldstein has . Both
of these are valid generating functions.)
1
1 2 2
, , ,
P
q q Q 1 1 2 2
''
F q P q Q
2
2
P q
1
1
q Q
2
2
p Q
1
1
P p
51
52. Canonical Transformation: Review
1
, ,
j
j
F
p q Q t
q
1
, ,
j
j
F
P q Q t
Q
1
F
K H
t
( , , ) ( , , ) , ,
j j j j
dF
p q H q p t P Q K Q P t old new t
dt
( , , )
( , , )
j j
j j
Q Q q p t
P P q p t
Type 1:
2
, ,
j
j
F
p q P t
q
2
, ,
j
j
F
Q q P t
P
2
F
K H
t
1( , , )
F F q Q t
Type 2:
2 ( , , ) i i
F F q P t Q P
3
, ,
j
j
F
q p Q t
p
3
, ,
j
j
F
P p Q t
Q
3
F
K H
t
Type 3:
3 ( , , ) i i
F F p Q t q p
4
, ,
j
j
F
q p P t
p
4
, ,
j
j
F
Q p P t
P
4
F
K H
t
Type 4:
4 ( , , ) i i i i
F F p P t q p Q P
52
53. Canonical Transformation: Review
- In general, one can use ANY one of the four types of generating
functions for the canonical transformation as long as the
transformation can be written in terms of the associated pairs of phase
space coordinates: (q, Q, t), (q, P, t), (q, Q, t), or (p, P, t).
- On the other hand, if the transformation is such that it cannot be written
in term of a particular pair: (q, Q, t), (q, P, t), (q, Q, t), or (p, P, t), then
that associated type of generating functions cannot be used.
- Generating function is useful as a bridge to link half of the original set
of coordinates (either q or p) to another half of the new set (either Q or P).
- the procedure in solving for F involves integrating the resulting partial
derivative relations from the CT condition
53