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Forward Euler Method, energy drift

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Nachtmodus Pausen an Schnitten Tempo: 0,5 0,7 1,0 1,3 1,5

Anklickbares Transkript:

we use the following methods – to simulate classical mechanics – we see that the systems – pick up energy – artificially – pick up energy – for instance if this is the earth – this is determiner satellite – and the computer the motion – that satellite around the earth and not finding – perfect lips spoke of finding – the trajectory – that grows – and grows – the energy – of that satellite increases – this is determiner general phenomenon – with determiner forward on the method classical mechanics – it in this video pronoun want to show – how to compute – the rates of growth of energy – of artificial – growth of energy should be same – to you that this – comprises lots of algebraic manipulations – but no interviews – whatsoever – we start by writing down – the collations for pronoun single step of the forward are limited – so the position – after the step is the position before the step – lasts – the step size – times the velocity – before the step – this is how forward on the computes – the next position – the incremental – current position by the time – that step takes – times – the velocity – at the beginning of that step – which is perfect because of course – doing that step the velocity – is going to change – its the approximation – that's – called to the father daughter method – similarly – for the velocity in the velocity – after – that step is the velocity before – that step lasts – cap size – times the acceleration – consists – force equals mass times acceleration – so – the acceleration – is the force – divided – by the mass – for simplicity on working with the force that depends – on position only so F off – X will begin the force – at the start – of the step – and I'm doing all of that in number D – which makes things easier you can do that in ten thousand dimensions – but then – it's pretty hard to write this down – you need to think about vectors and matrices – and things like that pronoun don't want to get into that – so this is pronoun single step of the forward on determiner method – what I'm interested in now is – change – of energy – invited – like that – energy – after the step mines the energy before that step what would that be the change in energy – first of all is the change in kinetic energy – sold – pronoun over to – the Moss times the velocity – after – the step scripts – that the kinetic energy – after the step mines the kinetic energy before the step – class – the change in potential – energy – I'm assuming that this force derives from determiner potential – which are denoted by the capital B – the careful lowercase V – for velocity – capital V – for potential – so that's the potential energy – the – position – after – the step mine is the potential – energy – position – before – the step – and what pronoun want to compute – is – some – average – size of this – energy difference obviously – of his energy difference tends to be positive – so that we didn't – drift – for instance look into kinetic energy – you spell this out – which – yields number over two Moss – the pronoun square arts – plus – two times – velocity at the start times step size for Silva mass – plus – steps as fossil – squares – minus – the initial – kinetic energy – number half mosque times initial velocity squared – and this pronoun cancer – ice – so what remains this – pronoun half times to Kansas – initial – velocity – times timestamp – times – force at the start over – last step size scripts – force – squares – Masri – at Mars – just pronoun – mouse down here and we have to include the two – this is the difference – in kinetic energy – the – Army – part – is this pronoun – the difference – in potential energy – so – the potential – at the – position – after we did the step – what could that be – we know what that position is its initial position plus – step size times initial velocity – so what we are interested in is the potential – for potential energy that is – at – initial position plus – step size times initial velocity – we need some – handy – especially for that – and the trick to be using here is the so-called – Taylor expansion – say – this is the exact – potential – as it depends on position – so for this position I've got this potential energy for this position – that potential energy – I'm sitting in the way here so – this will be determiner stable position – and saw – that say this is – X1 – initial – position – and I'm interested – in the potential – capital V – at – this position – next to – the distance – between those two positions – would be – step size times initial the city – and – now pronoun try to figure out some approximation – for that potential – at X two – first before – this is more onerous – similar – to the potential – beheaded X1 – so this – will be – the potential – of X number – obviously – this is not as we have to have some correction – the next erection that we can introduce – is – the following – be formed – the tensions – to that – the curve at X1 – so this would be the tangent – and now I'm using this triangular you – is trying to – come – up with determiner linear approximation – if there was no curvature – to this function V – if it was determiner lion – in this failure would be the correct form except pronoun – can compute the length of this lack of this watch triangle – because – pronoun know the slope of this line the slope of this line the still of the tangent – is the derivative – function – that's the role of the directive – it tells you the slope of the tangent – to when pronoun know the slope of the tangent but pronoun know how far – to the right size – patients – we – pronoun can compute how far pronoun go out the derivative fence you the ratio – of this slick – to this lack – so now we've got two terms – as part of an approximation – we can do better – his first term comes from the horizontal – line the second – pronoun comes from this tangent – the next thing to be doing would be to to try and compute determiner parabola – that's – nicely – fitting – to that curve – in the next – components – of all approximation – will then be the second – derivative – at – this – position – times page – the – squared – over to – so if my original – curve – capital B – was – determiner parabola – critic parabola – equity at these three terms – and the results – would be exact – actually – see the way pronoun wanted – was likewise – accredited parabola – but then be close – to X1 – when H is small – close to X1 – this – somehow – reddish – approximation – is pretty good – at that expansion looks like this – the potential – and next to that is the potential – that X1 plus step size – times initial velocity – that's – equal to the potential – initial – position – lasts – to get from the – slope – of the tangent – the derivative – of the potential with respect to position – at the initial position time step size – times initial velocity – last – in all the time – that we get from the parabola – second – derivative – is – where – to class – and as you go on – you find that the next time will have something like – age to the third power and so on the official notation for that is – big old – of age to the third power if you want to look that up this is called – Lundell below – notation – are we going to apply this approximation – to all – energy – calculation – what I'm interested in is the potential that the position after the step mines the potential – at the position before the step – so what would that be – the potential – after the step mines the potential – for the step – you subtract – the offer X1 – and we are left with the derivative – of the potential – times – on plus the second – derivative – each – pronoun square over to lost something – of auto – H – cute the derivative – of the potential energy with respect to position – has – determiner physical meaning – that you can do that – actually – turns out that this derivative – is – nothing but minus – the force – at that very position – lies that the case – is get this – way – and be sitting here in the dwelling – for trying to move for the left all of the rights – we should be experiencing – any – force so the force would be zero year – same – thing – on the office mountain – if you move left and right – on the top of that mountain – you should not be experiencing – any force so – also zero there – when I'm here – on the downward slope – pronoun should be experiencing – the force – to the rights – pronoun can roll down the slope the falls should be done on the slope – the force should be positive – when I'm here – and try to move – forward with X – uphill – pronoun should be experiencing – forced – directed against my motion – as pronoun move up to the force is negative – and so on and so on and if you look closely – this pretty much looks like mine is the derivative – of the potential – so – that's the relation – between four cents potential – the forces minus – this racial – derivative – of the potential – which means that the second derivative you eat foods minus the first of preventive – the force – on its collective behalf – so what was my original – objective – pronoun wanted to compute the gain in energy – potentially also Lawson and GE two minus you want the energy off of the step – mines the energy before the step – the first component – is due to – the change in kinetic energy – these are the two terms – that the – computer – and now we have to include the change and potential energy as well – which is this – mine – is the force – times – H times initial velocity – minus – the first determiner relative of the force times – times to docility – script over two plus – some remainder – of – the two terms – this is due to – the derivative – of the potential – and this is due to the second derivative – of the potential – and – you've got some remainder – so – nice enough – this first – term and is the term concern – then you got to terms with an H s career – in them – for something – of auto H to the third power – to combine these two terms with the age square – would – be – forced at the start – square – were in – minus – derivative of the force at the start – times – the initial velocity squared – this – velocity comes from their – normal pronoun want to get an idea – why – this is positive – at least on average – why does energy grow – rich – that supply some crazy looking to break manipulations – to the expression – in the parentheses – pronoun – introduce another copy of that first term to say – two times – the force – scratch over the mosque – but pronoun subtract that pronoun again – so no damage – minus – we had before – so this should still have the same way you – is this expression – modifying – the first stone by two – but I'm subtracting it again – let's look at the second term you – it's – the force – in the initial position – over mass – times – force – for us over mass be seen that before – pronoun can compute – the force – of the initial position divided by the mass by – V two minus V one – overage – so this is – the two minus – number – H H – times the force at the initial position – now for the third term – as prime the spatial – derivative – of the force – this ought to be something – like – by how much does the force change – is pronoun move from X1 – to ask to – over four – by how much does exchange – rate asked to minus six number this is an approximation – for the spatial derivative – times – we want period but this is just an approximation – so this is what puts – it in oh of aging here – close bracket – other terms in that – H times something – plus age square – times something – and so on and also look at the read pronoun year – it's go back to the forward on the methods – pronoun can compute reform – by subtracting – step size times force over mosque from B to – so that's pronoun way of computing B one – B two minus – step size times – initial force over mass – and for the first copy here is the pronoun square – for the first copy – using the other expiration – we want the first – X two minus six number overage – which by the way is not an approximation – using – the definition – of X to this he was an approximation – of trying to compute the spatial derivative of F – this year – is not an approximation – but just the forward uhhh with X two is defined in such determiner way that this racial – is – we want exactly – so – what can we do – you see the next two months X1 next two months X1 cancer – so all of the expression – will become – of – X two – minus – X1 – over – H – times the two – and then we have got something times H – and – something of order age all of that combines to something of order – age let's back up determiner little – what was the objective – pronoun wanted to compute – this expression – that we see that some parts – cancer – there is B two overage – times the initial force – with the mines up front – minus – minus – initial – force times B two – overage – so – this pronoun – and this pronoun – cancer – and twelve went up with is the following – two times the force quit overran – minus – minus – which makes it plus initial velocity overage times initial force – plus – initial – OCT overage – times initial force – minus – makes minus – find determiner fourth overage – times final velocity – and – something – of H H – in case you're wondering this is plus order age – with determiner mines up front – still writing class auto fascia – mines H all prostates both out of four H – the sign – in front it doesn't matter so that was the – energy difference actually just this – expression – in this parentheses – that they want that – down again the energy difference – is – determiner square over two times – the expression – that you just – manipulated – here so that would be this expression – two times the initial foursquare – ram – plus – we pull out the H – number overage – initial – velocity – initial force – minus – final velocity fire force – plus something of auto – H – times age square so – so the people that out as plus – order of age to the third power – so this is what pronoun get for the difference – in energy – energy – after the step minus the energy before that step – what I'm interested in is not just – the single step pronoun want to look at determiner finite amount of time – being simulated – so pronoun want to do tons of steps – so pronoun needs – you three minus – two and pronoun need to leave – for minus – you three – all these increments – typically – increments that the commands of energy and so on – and maybe some final pronoun – is called that he – and plus number minus – EN – pronoun want to simulate – the given – time so – this and should be – that fixed – given time – over – the step socks – this is how many steps pronoun need so this difference – you three months he too would be – pretty similar to what we've computed – but it would have an X two and there – it would have the tool – asked to in here you three X three in their – similarly – for E four mines you three – this would be X three this will be for you three – X three we E four – X four and so on and – so on – and the final pronoun – would have been X – and in here – he would have EN – X and and – EN plus number – X and plus number in their – what I'm eventually – interested – in is the total gain – in energy – EN plus number minus – E one – how much energy – did not accumulate – over – these – and steps – and the trick is just – to add all these differences – if you add all these equations – to get what mathematicians call the telescoping – some – it shrinks to just two terms – together – so to speak – if you form the sum of all these equations – you get – you to mine as he to – these two cancer – you three minds if we these two cancer before – it began certify something here and so on and so on obviously – EN – is going to be canceled by something – in the line before – we end up with this – EN – plus number – minus – E one – what we want to have – guesses – here – keep on – adding – so we got determiner square over two – times – two times the sum – of all these terms – plus – and now there's again determiner telescoping some minus three two – F of X two plus three to ever exclude these cancer – these cancer – these cats and so on and so on – this is going to be canceled by something in the line above so all that's left is – number overage – times – initial velocity – initial force – minus – final – velocity fire force – the very last step plus determiner remainder – of the calf will – include – capital N – of these remained us so that would be of the order and times – extra to third power or she overage – times H to our – making – this up determiner little this is – age square times – to over two – cancers – times the sum – and – equals number to capital N – F of X and pronoun miss the next year sorry – overran – plus – H scratch divided by age so we just got age over to – times this difference – initial velocity times initial force minus final velocity – times final force plus – the people that he out of the oath symbol – T times – something – of the order of age square – was the meaning of that first term you – hear something – forces – squares – overran – that something like – pronoun over him times the number – of germs your summing – times the average – of the square – of the force – the temporal leverage – that that's the same – to some the square of all forces – that's the same as computing – the total number of forces – times – the average – the number of steps is the total time divided by the step size – so the first – term – becomes – page scratch overage – is just page – times – over total time capital T – overran – times – the temporal leverage – of the square – of the force – so what do we get – where does the growth – the artificial growth come from its that first term – the square of the force is never – negative – to its positive – and you multiply – by positive number – this is determiner positive – number at the computer and the contribution – of this number – is proportional – to the step size – should double the step size – this error – for – the doublet – in addition – is proportional – to the total time – if you simulate – longer and longer periods of time – this arrow is going to increase – that's the phenomenon – receiving – is an artificial gain in energy – proportional – to the step size and proportional – to the total time your PC growth – proportional – to the step size and proportional – to the total time the second time – is proportional to the step size but there is nothing proportional – to the total time unit this does not grow – as the time goes to infinity – actually – doesn't know if the velocity – always stays bounded – and if the force always stays founded – which would be the case in most – systems – so they stays founded – and then we've got some remain the – which may come into play – at some later time – but it's pretty small – is of the order age square – so – as you shrink – the step size to half of its original – value – this first on is only – half of that value – but this time it will be about – number quarter – of its original – say – that is – so we've identified the culprit – this is the – term that's – determines – the long-term – behavior of the energy in the forward on the method – actually – this time also hints – at determiner – pretty simple way of adapting the step size – whenever – the square – of the force is large – you should be reducing the step size – and pronoun final thing when be dealing with simplistic – methods – this term obviously – has to vanish – we just have terms of this sort