In Lagrangian mechanics, the Lagrangian technique tells us that when dealing with particles or rigid bodies that can be treated as particles, the Lagrangian can be defined as: L = T-V where T is the kinetic energy of the particle, and V the potential energy of the particle. It is also advised to start with Cartesian coordinates when expressing the kinetic energy and potential energy components of the Lagrangian e.g. T = m (x² + y² + 2²). To express the kinetic energy and potential energy in some other coordinate system requires a set of transformation equations. 3.1 Taking into consideration the information given above, show that the Lagrangian for a pendulum of length 1, mass m, free to with angular displacement 0- i.e. angle between the string and the perpendicular is given by: 3.2 L=T-V=1²0² +mg | Cos Write down the Lagrange equation for a single generalised coordinate q. State name the number of generalised coordinates in problem 3.1. Hence write down the Lagrange equation of motion in terms of the identified generalised coordinates.
In Lagrangian mechanics, the Lagrangian technique tells us that when dealing with particles or rigid bodies that can be treated as particles, the Lagrangian can be defined as: L = T-V where T is the kinetic energy of the particle, and V the potential energy of the particle. It is also advised to start with Cartesian coordinates when expressing the kinetic energy and potential energy components of the Lagrangian e.g. T = m (x² + y² + 2²). To express the kinetic energy and potential energy in some other coordinate system requires a set of transformation equations. 3.1 Taking into consideration the information given above, show that the Lagrangian for a pendulum of length 1, mass m, free to with angular displacement 0- i.e. angle between the string and the perpendicular is given by: 3.2 L=T-V=1²0² +mg | Cos Write down the Lagrange equation for a single generalised coordinate q. State name the number of generalised coordinates in problem 3.1. Hence write down the Lagrange equation of motion in terms of the identified generalised coordinates.
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Transcribed Image Text:In Lagrangian mechanics, the Lagrangian technique tells us that when dealing with
particles or rigid bodies that can be treated as particles, the Lagrangian can be defined
as:
L = T-V where T is the kinetic energy of the particle, and V the potential energy of the
particle. It is also advised to start with Cartesian coordinates when expressing the
kinetic energy and potential energy components of the Lagrangian
e.g. T = m (x² + y² + 2²). To express the kinetic energy and potential energy in
some other coordinate system requires a set of transformation equations.
3.1 Taking into consideration the information given above, show that the Lagrangian
for a pendulum of length 1, mass m, free to with angular displacement - i.e.
angle between the string and the perpendicular is given by:
3.2
4.1
4.1.1
4.1.2
4.1.3
4.1.4
4.2
L = T-V = ²² +mg | Cos
Write down the Lagrange equation for a single generalised coordinate q.
State name the number of generalised coordinates in problem 3.1.
Hence write down the Lagrange equation of motion in terms of the identified
generalised coordinates.
What is meant by the following?
Frame of reference
Inertial frame of reference
Time dilation
Length contraction
State the postulates of the Special Theory of Relativity
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