# Lagrange Equations of Motion for a particle in a vessel

• Wombat11
In summary, the conversation discusses the substitution of variables rcos(Θ) and rsin(Θ) for x and y respectively, leading to the equation z=(b/2)r^2. The Lagrangian of the system is given as (1/2)m(rdot^2+r^2⋅Θdot^2+zdot^2)-mgz and the time derivative of z is found to be zdot=br⋅rdot. This is then plugged into the Lagrangian, resulting in (1/2)m( rdot^2 + r^2⋅Θdot^2 + b^2r^2⋅rdot^2) -
Wombat11
I start out by substituting rcos(Θ) and rsin(Θ) for x and y respectively. This gives me z=(b/2)r^2. The Lagrangian of this system is (1/2)m(rdot^2+r^2⋅Θdot^2+zdot^2)-mgz. (rdot and such is the time derivative of said variable). I then find the time derivative of z, giving me zdot=br⋅rdot and plug it into the Lagrangian giving me (1/2)m( rdot^2 + r^2⋅Θdot^2 + b^2r^2⋅rdot^2) - (1/2)mgbr^2. (plugged in regular z too). Finding the Lagrange equation of motion for 'r' gives me (ddot will be the second time derivative of said variable) 0= rddot(1+b^2r^2)-r⋅Θdot^2 + b^2⋅rdot^2⋅r + gbr . This is not correct though, the right answer has a negative term of b^2⋅rdot^2⋅r. Any help would be greatly appreciated.
Sorry about the notation, I have no idea how to put the equations into the computer.

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Welcome back in PF !

Wombat11 said:
Lagrange equation of motion
The ##\mathcal L## you work out (*)
$$(1/2)m( \dot r^2 + r^2\dot \theta^2 + b^2r^2\dot r^2) - (1/2)mgbr^2$$
looks good to me. What is the Euler-Lagrange equation you then use ?(*) I 'typed':
$$(1/2)m( \dot r^2 + r^2\dot \theta^2 + b^2r^2\dot r^2) - (1/2)mgbr^2$$

## 1. What are Lagrange Equations of Motion?

Lagrange Equations of Motion are a set of equations that describe the motion of a particle in a vessel based on the principle of least action. They are named after mathematician and physicist Joseph-Louis Lagrange.

## 2. How are Lagrange Equations of Motion derived?

Lagrange Equations of Motion are derived using the Lagrangian function, which takes into account the kinetic and potential energy of the particle in the vessel. The equations are derived by taking the partial derivatives of the Lagrangian with respect to the position and velocity of the particle.

## 3. What is the significance of Lagrange Equations of Motion?

Lagrange Equations of Motion are significant because they provide a more elegant and efficient way of describing the motion of a particle compared to Newton's laws of motion. They also take into account any constraints or forces acting on the particle, making them more versatile for complex systems.

## 4. Can Lagrange Equations of Motion be used for any type of vessel?

Yes, Lagrange Equations of Motion can be used for any type of vessel, as long as the vessel can be described by a Lagrangian function. This includes systems with multiple particles and complex geometries.

## 5. How are Lagrange Equations of Motion different from Newton's laws of motion?

While Newton's laws of motion describe the motion of a particle in terms of its position, velocity, and acceleration, Lagrange Equations of Motion describe the motion in terms of generalized coordinates and their corresponding velocities. This makes them more suitable for solving problems with multiple degrees of freedom and constraints.

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