Lagrangian Dynamics

Table of Contents

Notation

  • lagrangian_dynamics_0f53abab86f12437335845b64d601183f4a37cee.png position vector of particle of mass lagrangian_dynamics_fe52c58a96549528260fe0d7d04caf390dd47865.png
  • lagrangian_dynamics_d47d9fa7ecdce85a4637016a75e71770060a3d87.png is velocity
  • lagrangian_dynamics_f55f9cec4e23cbcccfd366786e2bf48076b5e494.png is linear momentum
  • lagrangian_dynamics_b3731d37cd447bd6c31809075a6be43b3d0b04ec.png is kinetic energy
  • lagrangian_dynamics_cc7920feedffd463b6cdcd819ffd84d2c0eec052.png is the unit vector perpendicular to lagrangian_dynamics_f83f5191363496f883ad2afe2d4dc32ce2b215e3.png (the unit position vector)

Stuff

Equations / Theorems

Galilean Transformation

Transforms from one inertial frame to another moving at constant velocity lagrangian_dynamics_79dc7b66c784a82e557ef3df794e82b67a292b17.png relative to it

lagrangian_dynamics_2563c3c751673d9df404b2e9550c5a2b320821f0.png

Then,

lagrangian_dynamics_502ec73ac97b3aee04ab4aefe38ab7f2b5f776cc.png

lagrangian_dynamics_193d9db838a35c6deb38ac2d28e627bdd9aed28b.png

Conservation laws

Linear momentum

Angular momentum

lagrangian_dynamics_d7ffb2ac6ef4186642b2a9cf59c1e619156b01a9.png

Proof

lagrangian_dynamics_c0627cc7c20ce2feabe5ace24e15c0943c791846.png

Remember, in this case we have lagrangian_dynamics_c610679dc9c49342166e220257829d6219ad4eae.png and thus $× lagrangian_dynamics_f7cf4b0d28ba269e3d11804d4b49e2aa51316062.png

Lagrange's equations

Equations of motion in terms of generalised coordinates.

lagrangian_dynamics_0c7e684906456abd9fff7603fdfcc7187c626249.png

  • equivalent to Newton's laws
  • constraint equation have been eliminated
  • constraint forces do not appear - they don't contribute to lagrangian_dynamics_d20840eb157e53d57057a3097891fe2c9a490000.png

Remember that the generalized forces lagrangian_dynamics_d20840eb157e53d57057a3097891fe2c9a490000.png are simply the forces projected onto the generalized coordinates lagrangian_dynamics_e733ea68c448561f4680420e3be96ceda74dda61.png.

Derivation

We restrict our attention to functions of the form lagrangian_dynamics_ef2c09221c7f48adf5e8eb7ab4d1e78c8dcf2d04.png, where lagrangian_dynamics_e230d2e8c21ed2d68abb4e10454eac2d59e60042.png denotes the generalised coordinates.

We assume Newton's Laws to be true in this derivation.

Since lagrangian_dynamics_f2ac5f4395f4e19ba80313daf86f05deb993b58d.png we can apply the "cancellation of dots" which just says that for the case where lagrangian_dynamics_cdd1cc131da6040eca078917132a377727053c44.png is not a function of lagrangian_dynamics_fc8a67b7ee79fa7dcf52c6aebe49847cc3b064b7.png, then we have

lagrangian_dynamics_6a39532a346c3b4a16acbc7732d6afa6eadfa45b.png

thus,

lagrangian_dynamics_4316e7b42a0bb9878324b96b0c8ed0e94734c2eb.png

lagrangian_dynamics_d64cdcbe4331bd8906ae8d1aa8b853e42f39db8d.png

Further,

lagrangian_dynamics_223f22369bb348e859953d1749d86fdb0672bdee.png

Summing over lagrangian_dynamics_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png

lagrangian_dynamics_0abcfd67508eb2cff6565ed8065a68b650fcf22e.png

we see that it's just the kinetic energy lagrangian_dynamics_b3731d37cd447bd6c31809075a6be43b3d0b04ec.png as a function of the generalised coordinates, velocities and time.

Thus,

lagrangian_dynamics_ace7c436db376f4097059a1cd2d2c590694c1151.png

where the first term is just lagrangian_dynamics_4a49c90f932bb21f3bb737e8f4f4b77fb3c976dd.png.

Since the constraint forces do now work, the sum over all constraint forces is zero.

Therefore we end up with Lagrange Equations

lagrangian_dynamics_6e09dfc9072fc8a8239adb68a776a0b3b1798bbc.png

Using conservative forces

If the forces are conservative there exists a potential energy function such that

lagrangian_dynamics_e5ba6691e6af8d478d7262040eef48ec241b1679.png

Assume $V = V(\{qj\}, t) then

lagrangian_dynamics_98395818419d056877e9a9dcb30f5d022baa3839.png

Since lagrangian_dynamics_7c34966129bf52d1a2efde929220a2443d5bc9c8.png, we may write

lagrangian_dynamics_bb11b51134c9109d670ce07c82ce2092f434786e.png

where lagrangian_dynamics_8d4220e91a4569589aaee819d7752a5ff1a26e7e.png

This is for holonomic constraints.

This applies to systems which also does not conserve their energy, unlike the usual

lagrangian_dynamics_1b979958196b6e29277c6cf31d50bbc2847be65a.png

which is only valid for systems which conserve energy.

Examples
  • Single particle in Cartesian Coordinate

    We have

    lagrangian_dynamics_37a8ca4e3027c81489066f9d00e38661bfb9e53a.png

    Substituting into Lagrange's equation for conservative forces

    lagrangian_dynamics_65db8b7d4d5ae05c42df1763ee6ae575b4f23bc8.png

  • Single particle in polar coordinates

    lagrangian_dynamics_9d3c1cadd9e2e016defb4b36fb23d1ea308b42f9.png

    lagrangian_dynamics_2f05366710d8cb9729ee29be08e807811b3c7768.png

    Substituing into Lagrange's equations wrt. lagrangian_dynamics_2dc980e89f64f5d0eb0fdb2ec4a66cb82b9c6f6b.png

    lagrangian_dynamics_c4440e628b23a6b44f44ceedab82ad2bde6e8fef.png

    and wrt. lagrangian_dynamics_e7a86163f39fa21d4a2ed66946369cdeb900ef42.png

    lagrangian_dynamics_6c64b3e4a625aeb9ae5748f01d7e242786f4d98f.png

Definitions

Intertial frame

A frame in which Newtons 1st and 2nd laws hold.

Newtons Laws

Newtons 3rd law

Weak

lagrangian_dynamics_78a0642398c573afa20bfeb7f21df49a4c0e8e63.png

for two objects lagrangian_dynamics_1f36e68578b771d42cf836dbdf0a6923a809fd5c.png and lagrangian_dynamics_311bf31f836cbf61de75e9461effe47dc0f184eb.png acting on each other.

Strong

Weak assumption AND lagrangian_dynamics_c004b71c50e75a846668213b3a448a403bbed79d.png acts along lagrangian_dynamics_609d80ea3f14c8350486546134399ed49a26c915.png

Effect potential

lagrangian_dynamics_cab6460fefea6229f73214d5746c627bd1fee49a.png

due to

lagrangian_dynamics_be9ed11b6df2543b9493a848ebdf830ad10bbce4.png

where lagrangian_dynamics_715b3ec0abcd3f3e6865484c05936212889bd462.png gives us

lagrangian_dynamics_aeeb5778b4d54baed014817c794af68c315e6b08.png

Constraint force

Constraint forces do no work in any small instantaneous displacement of the system consistent with the constraints themselves.

Does not mea that the constraint forces can do no work during the actual motion of the system, e.g. a particle constrained to lie on a surface which is itself moving: there may then be a component of the actual particle velocity in the direction of the constraint force, so that work is done.

Circular orbits

lagrangian_dynamics_2e77c8774ac7d8b27a48a1ddba20c077cda4eaf7.png

lagrangian_dynamics_2cf5b110d6c905a819b2ea119a88714a3fc4f0fd.png

Holonomic constraints

Example

lagrangian_dynamics_6144752331ca0c2e9a38e8cf25890adee97e5d30.png

i.e. it's a algebraic equation between the coordinates, not a differentiable relation and not an inequality.

Generalised coordinates

Consider a system with lagrangian_dynamics_a7c24c7f540c3d6c6effa114877324e1f8c7360c.png coordinates, i.e. 3D with lagrangian_dynamics_e10e2b430f95617381cdd6d6b52aed29fb971dff.png particles each with coordinates, and let these coordinates be denoted by $xi,$ where lagrangian_dynamics_ab5384e41341cebc9defe05736c995fea89d1a47.png.

That is, we're just "flattening" the lagrangian_dynamics_3910505d4a10db1b42ef864478fac4609df2c869.png matrix to a lagrangian_dynamics_a7c24c7f540c3d6c6effa114877324e1f8c7360c.png vector.

If lagrangian_dynamics_0b1128bc630dc6923120864befb3a99443664f83.png holonomic constraints, not all lagrangian_dynamics_e9f4d218474bc10aa94958ec30139aee865c0173.png are independent, and lagrangian_dynamics_cdfdb77f648a7baf135df2b59cfa0f4fb2d2280c.png set of independent variables

lagrangian_dynamics_787972282b8d67b945946712eb48efe801bc2a35.png

We might have dependence between the lagrangian_dynamics_e9f4d218474bc10aa94958ec30139aee865c0173.png due to the constraints being imposed, and thus representing the coordinates in the above way is just "removing" the dependence between the lagrangian_dynamics_e9f4d218474bc10aa94958ec30139aee865c0173.png.

Hence, we end up with a basis of lagrangian_dynamics_05251ce92fb1cc801b6b163557da28afde5d3039.png dimensions.

Our aim is to derive lagrangian_dynamics_6b1d98d1524f2609e864c1a5f53bb3edd2477588.png 2nd order differential eq. for the set of generalised coordinates lagrangian_dynamics_e230d2e8c21ed2d68abb4e10454eac2d59e60042.png.

Transformation from lagrangian_dynamics_2c64abb9fef7bb8531f60f7ec296ed03ff0227ad.png to lagrangian_dynamics_65430a0e4bd95c35c34235b0edcc71ad334c3f3a.png is invertible using constraint eq., that is:

lagrangian_dynamics_ec7336f7502d5cd58ca4315d48e3d43562e31191.png

lagrangian_dynamics_580a753f815a05ad0d6a713fa6d0a0c2187907f2.png cannot be varied independently without violating the constraints, whereas we can vary lagrangian_dynamics_e230d2e8c21ed2d68abb4e10454eac2d59e60042.png while still satisfying the constraints.

Generalised velocities

If lagrangian_dynamics_e230d2e8c21ed2d68abb4e10454eac2d59e60042.png denotes the generalised coordinates, then lagrangian_dynamics_fc8a67b7ee79fa7dcf52c6aebe49847cc3b064b7.png represents the generalised velocities.

Generalised forces

lagrangian_dynamics_2b7556437fd66c6ad3e2a52676f9abbfa15ddad9.png

Here lagrangian_dynamics_b27f43ceb9427a82a55805682f2f6dc50a2ca9b4.png may have a component in the direction of a constraint force, due to motion, and so the constraint force may do work, e.q. a body on a surface is utself moving.

"Instantaneous", i.e. lagrangian_dynamics_473eb94194cecf81f2acca79e0e18282725e542e.png

Then a small change consistent with constraints

lagrangian_dynamics_26b3ea45671df7a79295e847718f6a6f6849974f.png

In a virtual displacement the work done is

lagrangian_dynamics_98446ce9e2ef9acdafcf85df46f0234a537bf44d.png

since constraint forces do no work.

lagrangian_dynamics_e40073ed756acadf70882501ade66312c40a7dd3.png

where

lagrangian_dynamics_40468d3e05dda83059884a0a453b0431eb271687.png

Problems

Stuff