Lagrange's Theorem
•Download as PPT, PDF•
5 likes•5,791 views
The document discusses Lagrange's equations for describing the motion of particles and systems with constraints. It provides an example of using generalized coordinates to derive the equation of motion for a simple pendulum in terms of the angular coordinate φ. The Lagrangian approach eliminates constraint forces and allows problems to be solved in any coordinate system using Lagrange's equations.
Report
Share
Report
Share
![[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Problem 6.19 October 21, 2010 (x 1 ,y 1 ) (x 2 ,y 2 ) y(x) y x ds](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
Laplace transform
This document contains information about a Laplace transform topic presentation including:
- The names and enrollment numbers of 8 students working on the topic.
- The definition of the Laplace transform and some elementary functions transformed.
- Theorems on shifting, differentiation, integration, and multiplication of Laplace transforms.
- Examples of using Laplace transforms to evaluate integrals and find derivatives.
- The application of Laplace transforms to differential equations.
Chapter 2 laplace transform
The document provides an overview of the Laplace transform:
1. It introduces the Laplace transform and describes how it is used to transform functions from the time domain to the complex s-domain. This allows solving circuit problems involving initial conditions using algebraic equations rather than differential equations.
2. Key properties and theorems of the Laplace transform are described, including its use in solving linear time-invariant differential equations by taking the Laplace transform of both sides of the equation.
3. The inverse Laplace transform is explained as a way to transform signals back from the s-domain to the time domain. Common Laplace transform pairs and the Laplace transforms of basic circuit elements are also summarized.
Application of Laplace Transforme
The document provides an overview of topics related to Laplace transforms and their applications. It defines the Laplace transform and discusses some of its key properties, including linearity and how it relates to derivatives, integrals, and shifting theorems. The document also outlines how Laplace transforms can be used to evaluate integrals and solve differential equations. It provides examples of calculating the Laplace transforms of common functions.
Tensor analysis
1) The document discusses tensors and their properties, including tensors of rank one and two, contravariant and covariant tensors, and the metric tensor.
2) It explains how tensors transform under coordinate transformations and defines the gradient operator as a covariant vector.
3) The Minkowski metric is introduced as the metric tensor for flat spacetime in special relativity.
Laplace Transformation & Its Application
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.
Taylor series
The document discusses the history and development of Taylor series. Some key points:
1) Brook Taylor introduced the general method for constructing Taylor series in 1715, after which they are now named. Taylor series represent functions as infinite sums of terms calculated from derivatives at a single point.
2) Special cases of Taylor series, like the Maclaurin series centered at zero, were explored earlier by mathematicians like Madhava and James Gregory.
3) Taylor series allow functions to be approximated by polynomials and are useful in calculus for differentiation, integration, and approximating solutions to problems in physics.
Gauss jordan method.pptx
The document discusses the Gauss-Jordan elimination method for solving systems of linear equations. It involves transforming the augmented matrix of the system into reduced row echelon form using elementary row operations. The key steps are to turn the equations into an augmented matrix, use row operations to put the matrix in diagonal form with ones on the diagonal, and then divide elements to solve for the variables. An example problem demonstrates applying the Gauss-Jordan method to solve a system of three equations with three unknowns.
Ordinary differential equation
This document provides an introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.
Recommended
Laplace transform
This document contains information about a Laplace transform topic presentation including:
- The names and enrollment numbers of 8 students working on the topic.
- The definition of the Laplace transform and some elementary functions transformed.
- Theorems on shifting, differentiation, integration, and multiplication of Laplace transforms.
- Examples of using Laplace transforms to evaluate integrals and find derivatives.
- The application of Laplace transforms to differential equations.
Chapter 2 laplace transform
The document provides an overview of the Laplace transform:
1. It introduces the Laplace transform and describes how it is used to transform functions from the time domain to the complex s-domain. This allows solving circuit problems involving initial conditions using algebraic equations rather than differential equations.
2. Key properties and theorems of the Laplace transform are described, including its use in solving linear time-invariant differential equations by taking the Laplace transform of both sides of the equation.
3. The inverse Laplace transform is explained as a way to transform signals back from the s-domain to the time domain. Common Laplace transform pairs and the Laplace transforms of basic circuit elements are also summarized.
Application of Laplace Transforme
The document provides an overview of topics related to Laplace transforms and their applications. It defines the Laplace transform and discusses some of its key properties, including linearity and how it relates to derivatives, integrals, and shifting theorems. The document also outlines how Laplace transforms can be used to evaluate integrals and solve differential equations. It provides examples of calculating the Laplace transforms of common functions.
Tensor analysis
1) The document discusses tensors and their properties, including tensors of rank one and two, contravariant and covariant tensors, and the metric tensor.
2) It explains how tensors transform under coordinate transformations and defines the gradient operator as a covariant vector.
3) The Minkowski metric is introduced as the metric tensor for flat spacetime in special relativity.
Laplace Transformation & Its Application
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.
Taylor series
The document discusses the history and development of Taylor series. Some key points:
1) Brook Taylor introduced the general method for constructing Taylor series in 1715, after which they are now named. Taylor series represent functions as infinite sums of terms calculated from derivatives at a single point.
2) Special cases of Taylor series, like the Maclaurin series centered at zero, were explored earlier by mathematicians like Madhava and James Gregory.
3) Taylor series allow functions to be approximated by polynomials and are useful in calculus for differentiation, integration, and approximating solutions to problems in physics.
Gauss jordan method.pptx
The document discusses the Gauss-Jordan elimination method for solving systems of linear equations. It involves transforming the augmented matrix of the system into reduced row echelon form using elementary row operations. The key steps are to turn the equations into an augmented matrix, use row operations to put the matrix in diagonal form with ones on the diagonal, and then divide elements to solve for the variables. An example problem demonstrates applying the Gauss-Jordan method to solve a system of three equations with three unknowns.
Ordinary differential equation
This document provides an introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.
Second order homogeneous linear differential equations
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
Solution of System of Linear Equations
This document discusses methods for solving systems of linear equations. It covers direct methods like Gaussian elimination and LU factorization. Gaussian elimination reduces a system of equations to upper triangular form using elementary row operations. LU factorization expresses the coefficient matrix as the product of a lower triangular matrix and an upper triangular matrix. The document provides examples to demonstrate Gaussian elimination with partial pivoting and solving a system using LU factorization. Iterative methods are also introduced as an alternative to direct methods for large systems.
Laplace transform
The document presents information about Laplace transforms including:
- The definition and properties of the Laplace transform. It transforms differential equations into algebraic equations.
- Examples of applications in solving ordinary and partial differential equations, electrical circuits, and other fields.
- Limitations in that it can only be used to solve differential equations with known constants.
- Conclusion that the Laplace transform is a powerful mathematical tool used across many areas of science and engineering.
Analytic function
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
Presentation on laplace transforms
- The Laplace transform is a linear operator that transforms a function of time (f(t)) into a function of complex frequency (F(s)). It was developed from the work of mathematicians like Euler, Lagrange, and Laplace.
- The Laplace transform has many applications in fields like semiconductor mobility, wireless network call completion, vehicle vibration analysis, and modeling electric and magnetic fields. It allows transforming differential equations into algebraic equations that are easier to solve.
- For example, in semiconductors with varying material layers, the Laplace transform can relate the conductivity tensor to the Laplace transforms of electron and hole densities, enabling the determination of key properties like carrier concentration and mobility in each layer.
Laplace Transform and its applications
The Laplace transform is an integral transform that converts a function of time (often a function that represents a signal) into a function of complex frequency. It has various applications in engineering for solving differential equations and analyzing linear systems. The key aspect is that it converts differential operators into algebraic operations, allowing differential equations to be solved as algebraic equations. This makes the equations much easier to manipulate and solve compared to the original differential form.
Eigen value and eigen vector
This document provides information about eigenvalues and eigenvectors. It defines eigenvalues and eigenvectors as scalars (λ) and vectors (x) that satisfy the equation Ax = λx, where A is a matrix. It discusses properties of eigenvalues including that the sum of eigenvalues is the trace of A, and the product is the determinant. The characteristic equation is defined as det(A - λI) = 0, where the roots are the eigenvalues. Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation. Examples are given to demonstrate Cayley-Hamilton theorem.
Laplace transform: UNIT STEP FUNCTION, SECOND SHIFTING THEOREM, DIRAC DELTA F...
The document discusses the unit step function (also called the Heaviside function) and provides its definition and Laplace transform. It also discusses properties related to the Laplace transform of the unit step function, including:
1) The Laplace transform of the unit step function u(t-a) is 1/s when t ≥ a and 0 when t < a.
2) Using the shifting property, the Laplace transform of f(t)u(t-a) is e-asL[f(t+a)], where L[f(t)] is the Laplace transform of f(t).
3) An example calculates the Laplace transform of t2u(t-2) using the
Runge Kutta Method
This document summarizes the Runge-Kutta methods for solving differential equations numerically. It introduces the first, second, third, and fourth order Runge-Kutta methods and provides the equations for calculating each. An example of using the fourth order Runge-Kutta method to solve the differential equation dy/dx=x+y is shown step-by-step. The example calculates the solution to y(0.2) given y(0)=1 using increments of h=0.1.
EM3 mini project Laplace Transform
This document discusses the Laplace transform of periodic functions. It defines periodic functions as functions that repeat their values over a fixed time period. The Laplace transform can be used to change periodic functions into other functions in terms of the variable s. Specifically, the formula for the Laplace transform of a periodic function f(t) with period T is 1/(1-e-as) from 0 to T of e-st f(t) dt. The document proves this formula and provides examples of finding the Laplace transforms of periodic functions like sin(pt) and square waves. It concludes with some applications of Laplace transforms in engineering.
Tensor 1
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
First order linear differential equation
1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
Interpolation
This document discusses various interpolation methods used in numerical analysis and civil engineering. It describes Newton's divided difference interpolation polynomials which use higher order polynomials to fit additional data points. Lagrange interpolation polynomials are also covered, which avoid divided differences by reformulating Newton's method. The document provides examples of applying these techniques. It concludes with an overview of image interpolation theory, describing how the Radon transform maps spatial data to projections that can be reconstructed.
Using Laplace Transforms to Solve Differential Equations
This document discusses using Laplace transforms to solve differential equations. It begins with an introduction to Laplace transforms and their inventor Pierre-Simon Laplace. The author then explains that Laplace transforms can be used to transform differential equations into simpler algebraic equations that are easier to solve. As an example, the document walks through using Laplace transforms to solve a first order differential equation. It concludes that Laplace transforms are an important mathematical tool for solving differential equations.
Differential equations
- A differential equation relates an independent variable, dependent variable, and derivatives of the dependent variable with respect to the independent variable.
- The order of a differential equation is the order of the highest derivative, and the degree is the degree of the highest derivative.
- Differential equations can be classified based on their order (first order vs higher order) and linearity (linear vs nonlinear).
- The general solution of a differential equation contains arbitrary constants, while a particular solution gives specific values for those constants.
Differential Equations Lecture: Non-Homogeneous Linear Differential Equations
This document discusses the method of undetermined coefficients for solving nonhomogeneous second-order linear differential equations. It explains that the solution to a nonhomogeneous equation is the sum of the solution to the corresponding homogeneous equation and a particular solution to account for the nonhomogeneous term. It also outlines the steps to use the method of undetermined coefficients, which involves guessing a form for the particular solution based on the type of nonhomogeneous term and solving for the coefficients.
Cauchy integral theorem & formula (complex variable & numerical method )
1) The document discusses the Cauchy Integral Theorem and Formula. It states that if a function f(z) is analytic inside and on a closed curve C, then the integral of f(z) around C is equal to 0.
2) It provides examples of evaluating integrals using the Cauchy Integral Theorem when the singularities lie outside the closed curve C.
3) The Cauchy Integral Formula is introduced, which expresses the value of an analytic function F(a) inside C as a contour integral around C. Examples are worked out applying this formula to find the value and derivatives of functions at points inside C.
Poisson’s and Laplace’s Equation
This document discusses Poisson's and Laplace's equations in the context of a class on field theory taught by Prof. Supraja. It provides definitions and derivations of Poisson's and Laplace's equations in different coordinate systems. It gives examples of using these equations to find electric potential and field between charged surfaces, as well as the capacitance between them. It also states the uniqueness theorem, which guarantees a unique solution to Laplace's/Poisson's equation given the boundary conditions.
Application of analytic function
This document discusses the application of analytic functions to fluid flow, electrostatic fields, and heat flow problems. It explains that for incompressible fluid flow, the complex potential F(z) describes the flow, with its real part giving the velocity potential and imaginary part the stream function. It also describes how the electrostatic potential satisfies Laplace's equation and can be written as the real part of a complex potential. Finally, it explains that steady heat conduction problems are governed by Laplace's equation and the heat potential is the real part of a complex heat potential, with constant values representing isotherms and heat flow lines.
Laplace transform and its application
This document provides a summary of topics covered in an advanced engineering maths course. It includes definitions and properties of the Laplace transform, formulas for taking the Laplace transform of common functions, applications of Laplace transforms to solve differential equations, and examples of using Laplace transforms to evaluate integrals. The document lists 10 students enrolled in the course and 12 topics to be covered, such as the Laplace transform of derivatives and integrals, convolution theorem, and application to chemical engineering problems.
Two
Richard Feynman's high school physics teacher introduced him to the principle of least action, one of the most profound concepts in physics. The principle states that among all possible paths a physical system can take between two configurations, the actual path taken will be the one that minimizes the action. The action is defined as the time integral of the Lagrangian over the path, where the Lagrangian is the difference between the system's kinetic and potential energies. This principle allows physics to be formulated in terms of variational calculus and is the foundation for classical mechanics, electromagnetism, general relativity, and other physical theories.
Stephy index page no 1 to 25 2
This document discusses wave equations and their applications. It contains 6 chapters that cover basic results of wave equations, different types of wave equations like scalar and electromagnetic wave equations, and applications of wave equations. The introduction explains that the document contains details about different types of wave equations and their applications in sciences. Chapter 1 discusses basic concepts like relativistic and non-relativistic wave equations. Chapter 2 covers the scalar wave equation. Chapter 3 is about electromagnetic wave equations. Chapter 4 is on applications of wave equations.
More Related Content
What's hot
Second order homogeneous linear differential equations
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
Solution of System of Linear Equations
This document discusses methods for solving systems of linear equations. It covers direct methods like Gaussian elimination and LU factorization. Gaussian elimination reduces a system of equations to upper triangular form using elementary row operations. LU factorization expresses the coefficient matrix as the product of a lower triangular matrix and an upper triangular matrix. The document provides examples to demonstrate Gaussian elimination with partial pivoting and solving a system using LU factorization. Iterative methods are also introduced as an alternative to direct methods for large systems.
Laplace transform
The document presents information about Laplace transforms including:
- The definition and properties of the Laplace transform. It transforms differential equations into algebraic equations.
- Examples of applications in solving ordinary and partial differential equations, electrical circuits, and other fields.
- Limitations in that it can only be used to solve differential equations with known constants.
- Conclusion that the Laplace transform is a powerful mathematical tool used across many areas of science and engineering.
Analytic function
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
Presentation on laplace transforms
- The Laplace transform is a linear operator that transforms a function of time (f(t)) into a function of complex frequency (F(s)). It was developed from the work of mathematicians like Euler, Lagrange, and Laplace.
- The Laplace transform has many applications in fields like semiconductor mobility, wireless network call completion, vehicle vibration analysis, and modeling electric and magnetic fields. It allows transforming differential equations into algebraic equations that are easier to solve.
- For example, in semiconductors with varying material layers, the Laplace transform can relate the conductivity tensor to the Laplace transforms of electron and hole densities, enabling the determination of key properties like carrier concentration and mobility in each layer.
Laplace Transform and its applications
The Laplace transform is an integral transform that converts a function of time (often a function that represents a signal) into a function of complex frequency. It has various applications in engineering for solving differential equations and analyzing linear systems. The key aspect is that it converts differential operators into algebraic operations, allowing differential equations to be solved as algebraic equations. This makes the equations much easier to manipulate and solve compared to the original differential form.
Eigen value and eigen vector
This document provides information about eigenvalues and eigenvectors. It defines eigenvalues and eigenvectors as scalars (λ) and vectors (x) that satisfy the equation Ax = λx, where A is a matrix. It discusses properties of eigenvalues including that the sum of eigenvalues is the trace of A, and the product is the determinant. The characteristic equation is defined as det(A - λI) = 0, where the roots are the eigenvalues. Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation. Examples are given to demonstrate Cayley-Hamilton theorem.
Laplace transform: UNIT STEP FUNCTION, SECOND SHIFTING THEOREM, DIRAC DELTA F...
The document discusses the unit step function (also called the Heaviside function) and provides its definition and Laplace transform. It also discusses properties related to the Laplace transform of the unit step function, including:
1) The Laplace transform of the unit step function u(t-a) is 1/s when t ≥ a and 0 when t < a.
2) Using the shifting property, the Laplace transform of f(t)u(t-a) is e-asL[f(t+a)], where L[f(t)] is the Laplace transform of f(t).
3) An example calculates the Laplace transform of t2u(t-2) using the
Runge Kutta Method
This document summarizes the Runge-Kutta methods for solving differential equations numerically. It introduces the first, second, third, and fourth order Runge-Kutta methods and provides the equations for calculating each. An example of using the fourth order Runge-Kutta method to solve the differential equation dy/dx=x+y is shown step-by-step. The example calculates the solution to y(0.2) given y(0)=1 using increments of h=0.1.
EM3 mini project Laplace Transform
This document discusses the Laplace transform of periodic functions. It defines periodic functions as functions that repeat their values over a fixed time period. The Laplace transform can be used to change periodic functions into other functions in terms of the variable s. Specifically, the formula for the Laplace transform of a periodic function f(t) with period T is 1/(1-e-as) from 0 to T of e-st f(t) dt. The document proves this formula and provides examples of finding the Laplace transforms of periodic functions like sin(pt) and square waves. It concludes with some applications of Laplace transforms in engineering.
Tensor 1
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
First order linear differential equation
1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
Interpolation
This document discusses various interpolation methods used in numerical analysis and civil engineering. It describes Newton's divided difference interpolation polynomials which use higher order polynomials to fit additional data points. Lagrange interpolation polynomials are also covered, which avoid divided differences by reformulating Newton's method. The document provides examples of applying these techniques. It concludes with an overview of image interpolation theory, describing how the Radon transform maps spatial data to projections that can be reconstructed.
Using Laplace Transforms to Solve Differential Equations
This document discusses using Laplace transforms to solve differential equations. It begins with an introduction to Laplace transforms and their inventor Pierre-Simon Laplace. The author then explains that Laplace transforms can be used to transform differential equations into simpler algebraic equations that are easier to solve. As an example, the document walks through using Laplace transforms to solve a first order differential equation. It concludes that Laplace transforms are an important mathematical tool for solving differential equations.
Differential equations
- A differential equation relates an independent variable, dependent variable, and derivatives of the dependent variable with respect to the independent variable.
- The order of a differential equation is the order of the highest derivative, and the degree is the degree of the highest derivative.
- Differential equations can be classified based on their order (first order vs higher order) and linearity (linear vs nonlinear).
- The general solution of a differential equation contains arbitrary constants, while a particular solution gives specific values for those constants.
Differential Equations Lecture: Non-Homogeneous Linear Differential Equations
This document discusses the method of undetermined coefficients for solving nonhomogeneous second-order linear differential equations. It explains that the solution to a nonhomogeneous equation is the sum of the solution to the corresponding homogeneous equation and a particular solution to account for the nonhomogeneous term. It also outlines the steps to use the method of undetermined coefficients, which involves guessing a form for the particular solution based on the type of nonhomogeneous term and solving for the coefficients.
Cauchy integral theorem & formula (complex variable & numerical method )
1) The document discusses the Cauchy Integral Theorem and Formula. It states that if a function f(z) is analytic inside and on a closed curve C, then the integral of f(z) around C is equal to 0.
2) It provides examples of evaluating integrals using the Cauchy Integral Theorem when the singularities lie outside the closed curve C.
3) The Cauchy Integral Formula is introduced, which expresses the value of an analytic function F(a) inside C as a contour integral around C. Examples are worked out applying this formula to find the value and derivatives of functions at points inside C.
Poisson’s and Laplace’s Equation
This document discusses Poisson's and Laplace's equations in the context of a class on field theory taught by Prof. Supraja. It provides definitions and derivations of Poisson's and Laplace's equations in different coordinate systems. It gives examples of using these equations to find electric potential and field between charged surfaces, as well as the capacitance between them. It also states the uniqueness theorem, which guarantees a unique solution to Laplace's/Poisson's equation given the boundary conditions.
Application of analytic function
This document discusses the application of analytic functions to fluid flow, electrostatic fields, and heat flow problems. It explains that for incompressible fluid flow, the complex potential F(z) describes the flow, with its real part giving the velocity potential and imaginary part the stream function. It also describes how the electrostatic potential satisfies Laplace's equation and can be written as the real part of a complex potential. Finally, it explains that steady heat conduction problems are governed by Laplace's equation and the heat potential is the real part of a complex heat potential, with constant values representing isotherms and heat flow lines.
Laplace transform and its application
This document provides a summary of topics covered in an advanced engineering maths course. It includes definitions and properties of the Laplace transform, formulas for taking the Laplace transform of common functions, applications of Laplace transforms to solve differential equations, and examples of using Laplace transforms to evaluate integrals. The document lists 10 students enrolled in the course and 12 topics to be covered, such as the Laplace transform of derivatives and integrals, convolution theorem, and application to chemical engineering problems.
What's hot (20)
Second order homogeneous linear differential equations
Second order homogeneous linear differential equations
Laplace transform: UNIT STEP FUNCTION, SECOND SHIFTING THEOREM, DIRAC DELTA F...
Laplace transform: UNIT STEP FUNCTION, SECOND SHIFTING THEOREM, DIRAC DELTA F...
Using Laplace Transforms to Solve Differential Equations
Using Laplace Transforms to Solve Differential Equations
Differential Equations Lecture: Non-Homogeneous Linear Differential Equations
Differential Equations Lecture: Non-Homogeneous Linear Differential Equations
Cauchy integral theorem & formula (complex variable & numerical method )
Cauchy integral theorem & formula (complex variable & numerical method )
Similar to Lagrange's Theorem
Two
Richard Feynman's high school physics teacher introduced him to the principle of least action, one of the most profound concepts in physics. The principle states that among all possible paths a physical system can take between two configurations, the actual path taken will be the one that minimizes the action. The action is defined as the time integral of the Lagrangian over the path, where the Lagrangian is the difference between the system's kinetic and potential energies. This principle allows physics to be formulated in terms of variational calculus and is the foundation for classical mechanics, electromagnetism, general relativity, and other physical theories.
Stephy index page no 1 to 25 2
This document discusses wave equations and their applications. It contains 6 chapters that cover basic results of wave equations, different types of wave equations like scalar and electromagnetic wave equations, and applications of wave equations. The introduction explains that the document contains details about different types of wave equations and their applications in sciences. Chapter 1 discusses basic concepts like relativistic and non-relativistic wave equations. Chapter 2 covers the scalar wave equation. Chapter 3 is about electromagnetic wave equations. Chapter 4 is on applications of wave equations.
physics430_lecture11.ppt
This document summarizes a physics lecture on oscillations. It begins by reviewing Hooke's law and how it relates the force from a spring to displacement. It then shows that Hooke's law applies to small displacements from any equilibrium point using a Taylor series expansion. Simple harmonic motion is introduced as oscillatory motion governed by Hooke's law. The solutions to the differential equation for simple harmonic motion are derived and expressed in terms of sine and cosine functions. Examples are given of a mass on a spring and a bottle floating in water to illustrate simple harmonic oscillations. Energy considerations are also discussed showing how potential and kinetic energy oscillate out of phase during simple harmonic motion.
Small amplitude oscillations
This document summarizes classical dynamics and small amplitude oscillations. It discusses oscillatory motion near equilibrium positions and developing the theory using Lagrange's equations. Normal modes of coupled oscillating systems are explored, where the normal coordinates represent eigenvectors that oscillate at characteristic frequencies. The principles of superposition and matrix representations are used to analyze examples like two coupled pendulums and a system of two masses connected by three springs.
A General Relativity Primer
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
Lagrangian formulation 1
This document provides an overview of Lagrangian mechanics and constraints in classical mechanics. It defines different types of constraints including holonomic, non-holonomic, rheonomic, and scleronomic constraints. Generalized coordinates are introduced as a set of independent parameters that can describe the motion of a mechanical system with constraints. The configuration space is defined as a 3N-dimensional space where a point represents the configuration of a system of N particles. Constraints reduce the number of degrees of freedom from 3N coordinates to n generalized coordinates.
Lagrange
This document discusses Lagrangian dynamics and Hamilton's principle. It begins by introducing important notation conventions used in the chapter. It then provides an overview of Hamilton's principle and how it can be used to derive Lagrange's equations of motion. This allows problems to be solved in a general manner even when forces are difficult to express or some constraints exist. Examples are provided, including deriving the equation of motion for a simple pendulum using both Cartesian and cylindrical coordinates. The concept of generalized coordinates is also introduced to represent the degrees of freedom of a system.
What are free particles in quantum mechanics
detailed notes and solved problems on a free particle in quantum mechanics. For details visit
https://www.learnandenjoy-physics.com/
E27
This document summarizes elementary particles in physics. It describes how particles are classified into leptons and hadrons. Leptons include electrons, muons, taus and their neutrinos. Hadrons include baryons like protons and neutrons, and mesons. Interactions are also classified, including the electromagnetic, weak, and strong interactions. The electromagnetic interaction between charged leptons and photons is described based on local gauge invariance, resulting in a theory of quantum electrodynamics that agrees well with experiments.
Resume Mekanika 2 bab lagrangian - Fisika UNNES Nurul Faela Shufa
This document summarizes key concepts from a chapter on Lagrangian mechanics. It discusses the principle of least action and how it relates the Lagrangian to the difference between kinetic and potential energy. It then provides two examples - a pendulum attached to a movable support and a spherical pendulum - to illustrate applying Lagrange's equations of motion. It also discusses constraint forces and how Lagrange multipliers are used to account for constraints in variational problems.
physics430_lecture06. center of mass, angular momentum
This document summarizes a physics lecture on center of mass and angular momentum. It defines center of mass as a weighted average of particle positions, and shows that the center of mass of a system moves as if external forces act at that point. It also defines angular momentum for single and multiple particles, and shows that angular momentum is conserved for a system with no external torques. Key equations include definitions of center of mass, angular momentum, and the relationship between angular momentum and torque.
International Refereed Journal of Engineering and Science (IRJES)
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Mathematical formulation of inverse scattering and korteweg de vries equation
This document summarizes the mathematical formulation of inverse scattering and the Korteweg-de Vries (KdV) equation. It begins by defining inverse scattering as determining solutions to differential equations based on known asymptotic solutions, specifically by solving the Marchenko equation. It then discusses how the KdV equation describes shallow water waves and solitons, and how the inverse scattering transform method can be used to determine soliton solutions from arbitrary initial conditions. The document outlines the procedure, including deriving the scattering data from an initial potential function and using its time evolution to reconstruct solutions to the KdV equation at later times. It provides examples using reflectionless potentials, specifically obtaining the single-soliton solution from an initial sech^2
ALL THE LAGRANGIAN RELATIVE EQUILIBRIA OF THE CURVED 3-BODY PROBLEM HAVE EQUA...
We consider the 3-body problem in 3-dimensional spaces of nonzero
constant Gaussian curvature and study the relationship between the masses of
the Lagrangian relative equilibria, which are orbits that form a rigidly rotating
equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3-dimensional spaces of constant nonzero curvature: positive
elliptic and positive elliptic-elliptic, on 3-spheres, and negative elliptic, on hyperbolic 3-spheres. We prove that all these Lagrangian relative equilibria exist
only for equal values of the masses.
Cs jog
This document provides an introduction to the finite element method by first discussing the calculus of variations. It explains that the finite element formulation can be derived from a variational principle rather than an energy functional. It then presents three examples that illustrate functionals - the brachistochrone problem, geodesic problem, and isoperimetric problem. The document defines the concepts of extremal paths, varied paths, first variation, and the delta operator to derive the Euler-Lagrange equation, which provides the necessary condition for a functional to be extremized.
Planetary Motion- The simple Physics Behind the heavenly bodies
The document summarizes Kepler's laws of planetary motion and Newton's law of universal gravitation. It explains how Newton was able to derive Kepler's laws from his law of gravitation. It then discusses the central force problem and how to solve it to obtain planetary orbits. Finally, it briefly touches on limitations of the two-body approximation and possibilities like Lagrange points when more than two bodies are involved.
Magnetic monopoles and group theory.
This document discusses magnetic monopoles and solitons in field theory. It summarizes that solitons are finite-energy, non-dissipative solutions to classical wave equations that arise in non-linear theories. Magnetic monopoles can be constructed from potentials that have Dirac string singularities, requiring the Dirac quantization condition where magnetic charge is quantized. Several models are described where magnetic monopoles arise, including the 't Hooft-Polyakov model in 3+1 dimensions, where the mass of monopoles is related to the gauge boson mass. To date, no magnetic monopoles have been observed experimentally.
Monotonicity of Phaselocked Solutions in Chains and Arrays of Nearest-Neighbo...
This document summarizes a paper on phaselocked solutions in chains and arrays of coupled oscillators. It introduces the topic of coupled oscillators and their importance in modeling neural activity. It describes previous work analyzing one-dimensional chains of oscillators using continuum approximations. The current paper aims to rigorously prove the existence of phaselocked solutions in chains without requiring a continuum limit. It also analyzes two-dimensional arrays by decomposing them into independent one-dimensional problems under certain frequency distributions. Key results include proving monotonicity of phaselocked solutions and spontaneous formation of target patterns in two-dimensional arrays with isotropic synaptic coupling.
String theory basics
In tis slide, an introduction to string theory has been given. Apart from that, a simple proof of 26 dimensions of bosonic string theory is given (following Zwiebach's approach).
I explained this presentation in two parts (on my YouTube channel). Here are the links
_______________________________________________
Part 1
https://www.youtube.com/watch?v=QQA4JQ6Y-eo&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp
_______________________________________________
Part 2
https://www.youtube.com/watch?v=vhLCtLn79jE&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp&index=2
_______________________________________________
Ch6 central forces
Central forces are forces that point radially towards or away from a source and depend only on the distance from the source. For a central force, the angular momentum of a particle is conserved, and the particle's motion occurs in a single plane. By defining an effective potential energy, the two-dimensional problem can be reduced to an equivalent one-dimensional problem, simplifying the analysis. Solving the equations of motion yields the particle's position r and angle θ as functions of time t, or r as a function of θ, describing the trajectory.
Similar to Lagrange's Theorem (20)
Resume Mekanika 2 bab lagrangian - Fisika UNNES Nurul Faela Shufa
Resume Mekanika 2 bab lagrangian - Fisika UNNES Nurul Faela Shufa
physics430_lecture06. center of mass, angular momentum
physics430_lecture06. center of mass, angular momentum
International Refereed Journal of Engineering and Science (IRJES)
International Refereed Journal of Engineering and Science (IRJES)
Mathematical formulation of inverse scattering and korteweg de vries equation
Mathematical formulation of inverse scattering and korteweg de vries equation
ALL THE LAGRANGIAN RELATIVE EQUILIBRIA OF THE CURVED 3-BODY PROBLEM HAVE EQUA...
ALL THE LAGRANGIAN RELATIVE EQUILIBRIA OF THE CURVED 3-BODY PROBLEM HAVE EQUA...
Planetary Motion- The simple Physics Behind the heavenly bodies
Planetary Motion- The simple Physics Behind the heavenly bodies
Monotonicity of Phaselocked Solutions in Chains and Arrays of Nearest-Neighbo...
Monotonicity of Phaselocked Solutions in Chains and Arrays of Nearest-Neighbo...
Lagrange's Theorem
- 1. Physics 430: Lecture 15 Lagrange’s Equations Dale E. Gary NJIT Physics Department