Euler lagrange equation, mechanics,

• okkvlt
In summary: Such boundary conditions often lead to more accurate predictions of behavior than would be possible using Lagrange's equations alone.I read something about Noether's theorem which seems very interesting. Could somebody explain this theorem to me?Noether's theorem is a mathematical theorem stating that every algebraic function has a corresponding continuous function that is invariant under certain transformations.
okkvlt
Could somebody explain to me how lagrange multipliers works in finding extrema of constrained functions? also, what is calculus of variations and lagrangian mechanics, and can somebody explain to me what the lagrangian function is and the euler-lagrange equation. And, i read something about noethers theorem which seems very interesting. could somebody explain this theorem to me?

I came up with this proof of the law of conservation of energy. It makes sense to me. Is it right?
suppose f[x] is the force on an object with mass=1 in a force field that depends on position alone(is conservative).
f[x]=d2x/dt2

the definition of energy:
E=.5(dx/dt)^2-S[f]dx (here S is my symbol of integration)

because energy is constant the derivative of energy is zero.
dE/dt=0=.5d[(dx/dt)^2]/dt-d[S[f]dx]/dt

by the chain rule, .5d[(dx/dt)^2]/dt=(d2x/dt2)(dx/dt)
0=(d2x/dt2)(dx/dt)-d[S[f]dx]/dt

the next step is making the substitution d[S[f]dx]/dt=(d[S[f]dx]/dx)(dx/dt), which is possible because the dx above the dt and the dx under the integral cancel out leaving a dt below the integral.
0=(d2x/dt2)(dx/dt)-(d[S[f]dx]/dx)(dx/dt)

By the fundamental theorem of calculus, the differentation undoes the integration of f, giving
0=(d2x/dt2)(dx/dt)-f(dx/dt)
dividing through by dx/dt
0=(d2x/dt2)-f
f=(d2x/dt2)
which is the original equality. QED

Makes sense to me. the trickiest part for me was realizing that i could algebraicly manipulate the liebniz fraction d[S[f]dx]/dt=(d[S[f]dx]/dx)(dx/dt)=f*(dx/dt) to get rid of the integral.
I read another proof that uses lagrangians but i didnt understand it.
Could somebody explain what a lagrangian is to me

Lagrange Multipliers are a way to solve constrained min/max problems. Basically the concept is that for some function f and some constraint g, there will be a min/max such that the gradients of f and g are parallel.

In other words: $$\nabla f(x,y,z) = \lambda \nabla g(x,y,z)$$ where lambda is simply some scalar that makes f and g equal.

I don't really know that much about variational calculus, but the Lagrangian is simply a way to explain that dynamics of a system. In mechanics it's just simply a formulation of KE + PE = E

As for conservation of energy, you can't really prove it. It's an law that's based on experimental data. You're correct but you assumed that there is no energy lost in your derivation.

Feldoh said:
As for conservation of energy, you can't really prove it. It's an law that's based on experimental data. You're correct but you assumed that there is no energy lost in your derivation.

In a certain sense, sure, but one can show in the Lagrangian-based theory of classical mechanics that invariance with respect to time implies energy is conserved (or, perhaps more acurately, that there is a conserved quantity which we identity with the energy). So in that sense one can prove conservation of energy - to connect it to your claim, though, one must of course then prove that the world is indeed described by Lagrangian mechanics, which requires the experiments. =P

The Lagrangian function in dynamics is usually written as
L(q,qdot) = T(q,qdot) + V(q)
where
L = Lagrangian function
T = kinetic energy function for the system
V = potential energy function for the system

The calculus of variations is used to express Hamilton's Principle which is the physical principle from which the system equations are derived using the calculus of variations. Lagrange's equations simply short cut the process, going straight to the equations that result from the application of the calculus of variations to Hamilton's Principle. In most cases, this works just fine and nothing is lost. The equations of motion are developed in a very quick, efficient manner.

In a few circumstances, particularly in dealing with mixed systems (such as electromechanical systems, electroacoustic system, mechanical-hydraulic systems, discrete-continuous systems, etc), there are so-called "natural boundary conditions" that apply at the interface between the two systems which are not evident when Lagrange's equations are used but which come out naturally through the direct application of the calculus of variations to Hamilton's Principle.

1. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical tool used in mechanics to describe the motion of a system. It is derived from the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action functional.

2. How is the Euler-Lagrange equation used in mechanics?

The Euler-Lagrange equation is used to describe the dynamics of a mechanical system. It is a set of differential equations that relates the positions, velocities, and accelerations of particles in a system to the forces acting on them. This allows us to predict the future motion of a system based on its initial conditions.

3. What are the key assumptions made in the Euler-Lagrange equation?

The Euler-Lagrange equation is based on the assumption that the system is conservative, meaning that the total energy of the system is conserved. It also assumes that the system is holonomic, meaning that the constraints can be expressed as equations involving only the positions and velocities of the particles.

4. Can the Euler-Lagrange equation be applied to any system?

The Euler-Lagrange equation can be applied to any system that satisfies the assumptions mentioned above. However, it is most commonly used in classical mechanics to describe the motion of particles and rigid bodies.

5. How does the Euler-Lagrange equation relate to other equations in mechanics?

The Euler-Lagrange equation is closely related to other equations in mechanics, such as Newton's laws of motion and Hamilton's equations. In fact, Newton's laws can be derived from the Euler-Lagrange equation in certain cases, and Hamilton's equations can be obtained from the Euler-Lagrange equation using the Hamiltonian formalism.

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